F Ben Belgacem

F Jelassi
email: fjelassi@utc.fr

V Girault
email: girault@ann.jussieu.fr

Uniqueness' Failure for the Finite Element Cauchy-Poisson's Problem

Keywords: Kohn-Vogelius Duplication, Steklov-Poincaré Framework, Finite Element Method, Spurious modes, Discrete Holmgren principle, Meshes and Graphs

We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn-Vogelius artifice and the condensation Steklov-Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the discrete and exact problems behave in diametrically opposite ways. Indeed, existence of the discrete solution is always guaranteed while its uniqueness may be lost. In contrast, the solution of the exact problem may not exist, but it is unique. We show how existence of the so called "weak spurious modes", of the exact variational formulation, is source of instability and the reason why existence may fail. For the discrete problem, we find that the cause of non uniqueness is actually the occurrence of "spurious modes". We track their fading effect asymptotically when the mesh size tends to zero. In order to restore uniqueness, we recall the discrete version of the Holmgren principle, introduced in [Azaïez et al, IPSE, 18, 2011], and we discuss the effect on uniqueness of the finite element mesh, using some graph theory basic material.

Introduction

The inverse Cauchy-Poisson problem consists in finding a potential in a bounded domain, solution to the Poisson equation, and subjected simultaneously to Neumann and Dirichlet conditions on the "complete" part of the boundary (see [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF] and references therein). This pair of data, known as Cauchy's boundary conditions, are required to counter-balance the deficit of measures and information on the complementary "incomplete" part. For some reason, data are missing there and they ultimately need to be reconstructed using hopefully relevant numerical procedures. This data completion model is encountered in plenty of inverse industrial issues and engineering processes.

Non-invasive aiding tools in medical diagnosis or non-destructive techniques in industrial inspection

procedures are examples where solving a Cauchy-Poisson problem may be useful or even sometimes necessary. We refer for instance to [START_REF] Brühl | Crack detection using electrostatic measurements[END_REF][START_REF] Friedman | Determining cracks by boundary measurements[END_REF] for the latter and to [START_REF] Koshev | FEM-based scalp-to-cortex EEG data mapping via the solution of the Cauchy problem[END_REF][START_REF] Clerc | Cortical mapping by Laplace-Cauchy transmission using a boundary element method[END_REF] for the former. This is far from exhaustive; numerous references clearly reflect the variety of applications of Cauchy-Poisson's problem, and similar inverse boundary data problems.

The Cauchy-Poisson problem is severely ill-posed, as reported in 1953 in [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equation[END_REF] for a strip domain and fully proven in [START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF] in a general geometrical setting. This ill-posedness gives rise to difficult questions and poses a tough challenge to numerical computations. The variational frame we retain is the one found in [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]. It is derived as the Euler-Lagrange optimality condition for minimizing the Kohn-Vogelius duplication functional introduced in [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF]1985]. The resulting problem is then condensed on the incomplete boundary by means of the Steklov-Poincaré operators (see [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF]), leading to a variational system where the unknown is the very missing data. Before unfolding the details of our approach, let us say some words on the critical characteristics and properties of this ill-posed problem.

• Uniqueness is the good news; it is successfully established owing to the Holmgren principle extended to less regular differential equations by suitable Carleman estimates (see [START_REF] Tataru | A-priori estimates of Carleman's type in domains with boundary[END_REF]).

• Non-existence is a dominant feature of this inverse problem. Most often existence fails, unless Neumann and Dirichlet data are exact, which is unlikely.

• Third, and this point can be tied in a way to non-existence, the ill-posedness is severe, as can be seen by the deep sensitivity of the solution to tiny perturbations of Cauchy's boundary data (see [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equation[END_REF][START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF]). This brings serious concerns to the scientific computational community.

To readers interested in connected topics, we suggest the non-exhaustive list of publications and references therein [START_REF] Han | The finite element method in a family of improperly posed problems[END_REF][START_REF] Andrieux | On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries[END_REF][START_REF] Ben Belgacem | Analysis of Lavrentiev-finite element methods for data completion problems[END_REF][START_REF] Bourgeois | On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates[END_REF][START_REF] Feng | Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method[END_REF][START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF][START_REF] Rischette | Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data[END_REF][START_REF] Caubet | A dual approach to Kohn-Vogelius regularization applied to data completion problem[END_REF][START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF][START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF].

We are interested in fine aspects of the finite element approximation of the Cauchy-Poisson problem. When linear finite elements are used on a simplicial mesh and a Ritz-Galerkin procedure is followed, it was found that the discrete problem loses uniqueness, i.e., has a non zero kernel, but has at least one solution (see [START_REF] Azaïez | A Finite Element Model for the Data Completion Problem: Analysis and Assessment[END_REF][START_REF]Full discretization of Cauchy's problem by Lavrentiev-finite element method[END_REF]). It is unstable and handling its instability remains the major obstacle to numerical simulations. The main purpose of this work is to pinpoint the source of instability. We obtain the following results:

• First, the instability of the exact problem is caused by the presence of the so-called weak spurious modes, as defined in [START_REF] Vandeven | Compatibilité des espaces discrets pour l'approximation spectrale du problème de Stokes périodique/non périodique[END_REF]. They are oscillatory modes that are almost invisible by the variational form. They are responsible for the uniform coerciveness failure, even when the discrete Holmgren principle is valid.

• Next, the kernel of the finite element bilinear form may not be void ; functions in there are called discrete spurious modes and they break the uniqueness at the discrete level. We learn that they behave as weak spurious modes of the exact bilinear form, as the mesh size tends to zero. In other words, the finite element spurious modes become hidden weak spurious modes of the continuous problem, for small mesh sizes. The nice point here is that these spurious modes are visible; they belong to the kernel of (the stiffness matrix of) the system. Most iterative solvers are able to filter them out.

• Finally, we investigate a class of meshes for possible cures. We recall the discrete version of the Holmgren principle, and illustrate its capacity to restore uniqueness of the discrete solution.

We exhibit the influence of the mesh on the validation of the Holmgren criterion and give some necessary conditions for its validity (in two dimensions for the sake of simplicity). In the general case, we show how Lavrentiev's regularization enables the discrete missing data to eliminate the spurious modes. This enhances considerably its relevance.

The content is as follows. In Section 2, the derivation of the variational formulation is recalled and existence of weak spurious modes is shown in Section 3. Section 4 is dedicated to the discretization of the variational problem by linear finite elements on a simplicial grid. The discrete spurious modes are studied in Section 5 and conflicting results with the continuous setting are discussed. In Section 6, the discrete Holmgren principle is recalled as well as its capacity to restore uniqueness of the discrete solution. By means of some graph theory, we show in Section 7, how the mesh may enforce the validity of this criterion. In Section 8, we establish the beneficial effect of Lavrentiev's regularization on the discrete solution. Section 9 is devoted to some simple numerical illustrations confirming the predictions drawn from the theoretical study in this work.

Preliminary Notations-Let x denote the generic point of a bounded domain Ω of R d , d = 2, 3.

The Lebesgue space of square integrable functions L 2 (Ω) is endowed with the natural inner product

(•, •) L 2 (Ω) ; the associated norm is • L 2 (Ω) . The Sobolev space H 1 (Ω) is H 1 (Ω) = v ∈ L 2 (Ω) : ∇ v ∈ L 2 (Ω) d .
It is provided with the norm • H 1 (Ω) , and the semi-norm is denoted by

| • | H 1 (Ω) . Let Υ ⊂ ∂Ω
be a connected component of the boundary. The space H 1/2 (Υ) is the set of the traces over Υ of all the functions of H 1 (Ω) and we adopt the notation H -1/2 (Υ) for the dual space of H 1/2 (Υ), the duality between the two spaces being denoted by •, • H 1/2 (Υ) . We refer to [START_REF] Adams | Sobolev Spaces[END_REF] for a detailed study of fractional Sobolev spaces.

θ Ω

Γ I Γ C Figure 1: Γ C is accessible to measures and Γ I is out of reach
No data can be measured on the boundary Γ I ; it may be unreachable. To counter-balance the lack of information on Γ I , we apply a flux across Γ C and collect measurements of the potential on the same Γ C , so that we have there a pair of Cauchy's conditions. The resulting model problem reads therefore, as:

find u ∈ H 1 (Ω) solution of -div(a∇ u) = 0 in Ω, u = g, a∂ n u = ϕ on Γ C , u = ? on Γ I , (1)
where the conduction parameter a(•) is a given function in W 1,∞ (Ω) such that a(•) ≥ a * > 0. The pair of Cauchy's boundary conditions (g, ϕ) are given in

H 1/2 (Γ C ) × H -1/2 (Γ C ).
We use below the following notation: H 1 I (Ω) stands for the subspace of H 1 (Ω) defined by :

H 1 I (Ω) = v ∈ H 1 (Ω) : v |Γ I = 0 .
The subspace

H 1 C (Ω) is defined similarly with v |Γ I = 0 replaced by v |Γ C = 0. We have thence in particular H 1 0 (Ω) = H 1 I (Ω) ∩ H 1 C (Ω).
Following [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF], we work with a variational formulation based on the Kohn-Vogelius duplication argument [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF], complemented with Dirichlet-to-Neumann operators (see [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF]). The single unknown of the problem is the very missing boundary data on Γ I .

To trigger the duplication argument, we consider two auxiliary basic functions, u D (µ, g), solution of a Laplace problem with Dirichlet boundary conditions,

-div(a∇u D (µ, g)) = 0 in Ω, u D (µ, g) = g on Γ C , u D (µ, g) = µ on Γ I , (2)
and u N (µ, ϕ), solution of a Laplace problem with mixed Dirichlet-Neumann boundary conditions,

-div(a∇u N (µ, ϕ)) = 0 in Ω, a∂ n u N (µ, ϕ) = ϕ on Γ C , u N (µ, ϕ) = µ on Γ I . (3)
It can be shown (see [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]), that Problem (1) has a solution if and only if there exists λ in H 1/2 (Γ I ) that preserves the fluxes on Γ I , more precisely,

a∂ n u D (λ, g) = a∂ n u N (λ, ϕ), on Γ I . (4)
Writing ( 4) in variational form requires some additional notation. Let

u N (µ) := u N (µ, 0), u D (µ) := u D (µ, 0), ȗN (ϕ) := u N (0, ϕ), ȗD (g) := u D (0, g).
Consider now the bilinear forms,

s(•, •), s D (•, •), s N (•, •) : ∀χ, µ ∈ H 1/2 (Γ I ), s(χ, µ) = s D (χ, µ) -s N (χ, µ) = Ω a∇ u D (χ) • ∇ u D (µ) dx - Ω a∇ u N (χ) • ∇ u N (µ) dx, (5)
and the linear forms

ℓ(•), ℓ D (•), ℓ N (•), ∀µ ∈ H 1/2 (Γ I ), ℓ(µ) = ℓ D (µ) -ℓ N (µ) = - Ω a∇ ȗD (g) • ∇ u D (µ) dx -ϕ, u N (µ) H 1/2 (Γ C ) .
The variational formulation of Cauchy's problem consists in :

finding λ ∈ H 1/2 (Γ I ) such that s(λ, µ) = ℓ(µ), ∀µ ∈ H 1/2 (Γ I ). (6)
The equivalence of ( 1) and ( 6) is stated in [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]. The milestone of the analysis proposed here is the ill-posedeness of [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF]. It is pointed out in [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equation[END_REF], through an example, and fully proven in [START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF], in a general context.

In this work, we will repeatedly use the notation

|v| a,H 1 (Ω) = √ a∇v L 2 (Ω) d .
The mapping

µ → s D (µ, µ) = |u D (µ)| a,H 1 (Ω) ,
is equivalent to the norm of H 1/2 (Γ I ), see [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF]. It is denoted • s D and we shall use it as the Hilbert norm of H 1/2 (Γ I ).

Remark 2.1 It can be checked that the bilinear form s(•, •)

satisfies ∀µ ∈ H 1/2 (Γ I ), s(µ, µ) = |u N (µ) -u D (µ)| 2 a,H 1 (Ω) . (7)
Owing to the symmetry of s(•, •), problem (6) may be equivalently expressed by minimizing the energy functional J(•),

J(µ) = 1 2 s(µ, µ) -ℓ(µ). (8)
In fact, J(•) coincides with the Kohn-Vogelius functional, modulo an additive constant. Indeed, an easy computation gives,

J(µ) = 1 2 |u D (µ, g) -u N (µ, ϕ)| 2 a,H 1 (Ω) - 1 2 |ȗ D (g) -ȗN (ϕ)| 2 a,H 1 (Ω) . (9)
This result stems in particular (see [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]) from the fact that

ϕ, u N (µ) H 1/2 (Γ C ) = Ω a∇ ȗN (ϕ) • ∇ u N (µ) -u D (µ) dx. ( 10
)
As a result, the variational problem (6) turns out to be the Euler-Lagrange equation related to the minimization of J(•).

Variational Spurious Modes

The bilinear form s(•, •), defined in [START_REF] Azaïez | On Cauchy's Problem. II. Completion, Regularization and Approximation[END_REF], is continuous and symmetric. Owing to [START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF], it is positive definite (see [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]). Hence it defines a norm on H 1/2 (Γ I ),

∀µ ∈ H 1/2 (Γ I ), µ s = s(µ, µ).

But, as will be shown in Lemma 3.3 below, this norm cannot be equivalent to the norm

• H 1/2 (Γ I ) ,
and hence the bilinear form s(•, •) is not elliptic. This lack of ellipticity causes inexistence of solutions for some data; they are called inexact data.

We shall explore further this issue and find that it is triggered by the presence of Weak Spurious

Modes. Before writing a precise definition of "spurious modes" and illustrating their existence, let us recall two interesting results based on the Holmgren uniqueness theorem (see [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF]Theorem 3.3.1]).

Lemma 3.1 Let g be given in H 1/2 (Γ C ).
The subset of the Neumann data ϕ for which problem (6) has a solution, is dense in

H -1/2 (Γ C ).
The complementary subset of ϕ for which existence for (6) fails is also dense in H -1/2 (Γ C ).

Proof: The proof of the first result stems from the Holmgren uniqueness principle and can be found for instance in [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]Proposition 2.3].

The proof of the second one proceeds by contradiction. Assume the result is false.

-Let us start with g = 0. By assumption, there exists an open ball in H -1/2 (Γ C ) that contains data ϕ that are compatible with g = 0, thus the pair (g = 0, ϕ) is exact. Making a translation to center that ball at the origin, using a dilation to expand its radius to infinity, and relying on the linearity of problem (6), we conclude that the whole space H -1/2 (Γ C ) is composed of fluxes ϕ so that (g = 0, ϕ) are exact data.

-Now, let g be arbitrary. It can be seen, once again by linearity, that the pair (g, ϕ) is exact if and only if (0, ϕ -a∂ n ȗD (g)) is also exact. By the above argument, we deduce that any flux data ϕ is compatible with g. This is of course not true. The proof is complete.

Remark 3.1

The following counterpart of this density result is also true: let ϕ ∈ H -1/2 (Γ C ) be given.

The subset of the Dirichlet data g so that (g, ϕ) are exact, is dense in

H 1/2 (Γ C ).
The complementary subset of g so that (g, ϕ) are inexact, is also dense in

H 1/2 (Γ C ).
It is noteworthy that the whole set of exact data (g, ϕ) is a subspace of H 1/2 (Γ C ) × H -1/2 (Γ C ); the complementary subset of inexact data is not.

A direct consequence of the density of exact data is the following informative result, also established in [8, Proposition 3.5] Lemma 3.2 We have that: for any given data (g, ϕ),

inf λ∈H 1/2 (Γ I ) sup µ∈H 1/2 (Γ I ) s(λ, µ) -ℓ(µ) µ s D = 0. ( 11
)
If the pair (g, ϕ) is exact, the infimum is realized and the argument that realizes the minimum in [START_REF]Full discretization of Cauchy's problem by Lavrentiev-finite element method[END_REF] is the unique solution of (6).

Remark 3.2 It is also possible to prove that inf

λ∈H 1/2 (Γ I ) sup µ∈H 1/2 (Γ I ) s(λ, µ) -ℓ(µ) µ s = 0,
a stronger result than [START_REF]Full discretization of Cauchy's problem by Lavrentiev-finite element method[END_REF], since the strong norm µ s D is replaced by the weak norm µ s . The result can be inferred following the lines in the proof of [START_REF] Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF]Theorem 4.4].

Our object now is to gain a further insight on the cause of ill-posedness of Problem [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF], and on the consequences of inexact data. It turns out that inexactness is produced by some "weak spurious modes" suffered by the bilinear form s(•, •). This is stated in

Lemma 3.3 There exists a sequence (ξ n ) n≥0 ⊂ H 1/2 (Γ I ) such that lim n→∞ ξ n s D = +∞ and lim n→∞ ξ n s = 0. ( 12
)
Proof: The proof can be done by means of the open map theorem. However, we prefer a more constructive proof based on Lemma 3.2.

It proceeds in two steps, first the construction of (ξ n ) n≥0 satisfying the last statement in [START_REF] Bourgeois | On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates[END_REF], and next the unboundedness of this sequence in • s D .

-Let (g, ϕ) be a pair of inexact data. Then, owing to Lemma 3.1, there exists a sequence (ϕ n ) n≥0

that converges in H -1/2 (Γ C ) to ϕ, such that the pairs (g, ϕ n ) are exact. Denote by z n the solution of [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF] with boundary data (g, ϕ n ) on Γ C . A simple subtraction yields that

s(z n -z m , µ) = -ϕ n -ϕ m , u N (µ) 1/2,Γ C = -ϕ n -ϕ m , u N (µ) -u D (µ) 1/2,Γ C , ∀µ ∈ H 1/2 (Γ I ), since u D (µ) = 0 on Γ C . Setting ζ (n,m) = z n -z m , we derive that s(ζ (n,m) , µ) ≤ ϕ n -ϕ m H -1/2 (Γ C ) u N (µ) -u D (µ) H 1/2 (Γ C ) , ∀µ ∈ H 1/2 (Γ I ).
By the trace theorem, and owing to [START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF], we obtain that

s(ζ (n,m) , µ) ≤ C ϕ n -ϕ m H -1/2 (Γ C ) |u N (µ) -u D (µ)| a,H 1 (Ω) ≤ C ϕ n -ϕ m H -1/2 (Γ C ) µ s , ∀µ ∈ H 1/2 (Γ I ). ( 13
)
Taking µ = ζ (n,m) , passing to the limit as (n, m) → +∞( 1) shows that ζ (n,m) satisfies the second part of ( 12).

-It remains to check that (ζ (n,m) ) n,m≥0 tends to infinity in the norm • s D . We argue by contradiction: assuming that it is bounded (with respect to • s D ), we will prove that its weak limit is zero in H 1/2 (Γ I ). This will result in the fact that the pair (g, ϕ) is exact, contradicting the assumption.

Assume that (ζ (n,m) ) n,m≥0 is bounded in • s D . We refer to the lectures in [START_REF] Habil | Double sequences and double series[END_REF] (see also references therein), for handling the doubly indexed sequence. According to this reference, there exists a subsequence (ζ (n k m k ) ) k≥0 , with (n k , m k ) tending to infinity with k, that converges weakly to a [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Theorem 3.18]). As s(•, •) is continuous on H 1/2 (Γ I ), we can pass to the limit in (13) and obtain

function ζ in H 1/2 (Γ I ) as k → ∞ (see
s(ζ, µ) = 0, ∀µ ∈ H 1/2 (Γ I ).
Given that s(•, •) is positive definite, we get necessarily ζ = 0. By the way, this implies also that zero is the only "weak" limit point of the bounded sequence (ζ (n,m) ) n,m≥0 . Recalling that

ζ (n,m) = z n -z m , we have lim n,m→+∞ z n -z m = 0, weakly in H 1/2 (Γ I ),
(see [START_REF] Habil | Double sequences and double series[END_REF]Theorem 6.8]). This implies in particular that lim n,m→+∞

s D (z n -z m , µ) = 0, ∀µ ∈ H 1/2 (Γ I ),
and thus the sequence (z n ) n≥0 fulfills the Cauchy criterion with respect to the weak topology.

Following Lemma 11.1 in the appendix, it has a weak limit z ∈ H 1/2 (Γ I ). Now, we pass to the limit in the variational formulation [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF] of the Cauchy problem with the data (g, ϕ n ) and the solution z n , and find that the limit z solves (6) with data (g, ϕ), contradicting the first assumption, that is (g, ϕ) are inexact. This ends the proof.

Remark 3.3 The sequences (ξ n ) n≥0 ⊂ H 1/2 (Γ I ) constructed in the proof of Lemma 3.3 are called "weak spurious modes". Though this concept occurs in several fields, it has been explicitly introduced by H. Vandeven in the community of spectral methods (see [START_REF] Vandeven | Compatibilité des espaces discrets pour l'approximation spectrale du problème de Stokes périodique/non périodique[END_REF]) and we follow precisely his definition. These modes are highly oscillatory. That is why their strong norm ( ξ n s D ) n≥0 blows up while they are dampened by the weak norm ( ξ n s ) n≥0 . They are certainly excited by inexact data (g, ϕ). Even worse, they are able to destabilize the solution λ for the exact data.

As seen from the proof above, their action is accompanied by an amplification of their strong norms, while their effect on the consistency of ( 6) is fading. That is expected to happen during numerical simulations.

Remark 3.4 Sometimes, it may be useful to normalize these modes, i.e. set ξ n s D = 1, while their weak norm ξ n s still tends to 0. They will be called "normalized weak spurious modes". By making the change in variable ξ n := ξn ( ξn s) 1-ǫ , for a small ǫ, we can reconstruct the blowing-up weak spurious modes. This may also give an indication on their rate of convergence to infinity, most probably exponentially fast (see [START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF]).

Finite Element Approximation

We propose a finite element approximation of problem [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF]. Let h > 0 be the discretization parameter. The symbol T h stands for a regular family of triangular or tetrahedral partition of Ω made of triangular or tetrahedral elements with maximum mesh size h,

Ω = T ∈T h T.
The family T h is regular and thus does not accept degenerate triangles or tetrahedra (see [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]). Let P 1 be the space of polynomials in two or three variables according to the dimension, with total degree at most one. We define X h to be the basic finite element space,

X h = v h ∈ C ( Ω) ; ∀T ∈ T h , v h|T ∈ P 1 .
We shall mostly use the subspace X I,h = X h ∩ H 1 I (Ω). Its alter ego subspace, X C,h , is defined in the same way. Then, we set

X 0,h = X I,h ∩ X C,h = X h ∩ H 1 0 (Ω).
The missing boundary condition λ is approximated in H I,h , the trace space on Γ I of X h , defined as

H I,h = µ h = v h | Γ I ; v h ∈ X h . (14)
Similarly, the symbol H C,h is the subspace inherited from X h by the trace operator. They are both standard finite element spaces built on the trace of the mesh T h , on each boundary, Γ I for H I,h

and Γ C for H C,h . Thus, they are subspaces of H 1/2 (Γ I ) or H 1/2 (Γ C ) according to the context.

The Cauchy data are discretized in H C,h . They are denoted g h and ϕ h , some good approximations of the exact data (g, ϕ). Note that although ϕ belongs to H -1/2 (Γ C ), we choose, for the sake of simplicity, to approximate it by the smoother functions of H C,h . This choice has no influence on the analysis of ill posedness. Then, for all µ h in H I,h , the function

u D,h (µ h , g h ) ∈ X h solves Ω a∇ u D,h (µ h , g h ) • ∇ v h dx = 0 ∀v h ∈ X 0,h , u D,h (µ h , g h ) = µ h on Γ I , u D,h (µ h , g h ) = g h on Γ C , (15)
and

u N,h (µ h , ϕ h ) ∈ X h solves Ω a∇ u N,h (µ h , ϕ h ) • ∇ v h dx = Γ C ϕ h v h d Γ ∀v h ∈ X I,h , u N,h (µ h , ϕ h ) = µ h on Γ I . (16)
Both systems [START_REF] Brown | GMRES on (nearly) singular systems[END_REF] and ( 16) are well posed. Each one has a unique solution. Similarly, the solutions u D,h (µ h ), u N,h (µ h ), ȗD,h (g h ) and ȗN,h (ϕ h ) are denoted by analogy to the exact situation of Section 2.

With these discrete functions, we define the discrete bilinear forms

s h (•, •), s D,h (•, •), and s N,h (•, •), ∀χ h , µ h ∈ H I,h , s h (χ h , µ h ) = s D,h (χ h , µ h ) -s N,h (χ h , µ h ), = Ω a∇ u D,h (χ h ) • ∇ u D,h (µ h ) dx - Ω a∇ u N,h (χ h ) • ∇ u N,h (µ h ) dx.
Likewise, the form ℓ(•), is discretized as

ℓ h (µ h ) = - Ω a∇ ȗD,h (g h ) • ∇ u D,h (µ h ) dx - Γ C ϕ h u N,h (µ h ) dΓ, ∀µ h ∈ H I,h .
The finite element discretization of problem ( 6) is :

find λ h ∈ H I,h such that, s h (λ h , µ h ) = ℓ h (µ h ), ∀µ h ∈ H I,h . (17)
Next, let

µ h s D,h = s D,h (µ h , µ h ) = |u D,h (µ h )| a,H 1 (Ω) .
In [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF], it is established that • s D,h is a norm on H I,h uniformly equivalent to • s D , more precisely,

∀µ h ∈ H I,h , µ h s D,h ≤ C µ h s D , µ h s D ≤ µ h s D,h , (18)
with a constant C independent of h. Regarding s h (•, •), we have as in the exact situation (see [5, Lemma 5.2]),

∀µ h ∈ H I,h , s h (µ h , µ h ) = |u D,h (µ h ) -u N,h (µ h )| 2 a,H 1 (Ω) . (19)
This allows to define the semi-norm

µ h s h = s h (µ h , µ h ),
that is not necessarily a norm on H I,h , i.e., s h (µ h , µ h ) = 0, does not necessarily imply that µ h = 0, as we shall see below.

As will be illustrated in the sequel, the amazing effect of the finite element approximation is that existence and uniqueness are literally reversed compared to the continuous problem. Existence fails for (6) while it is guaranteed for [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF]. Uniqueness is valid for ( 6) and most often breaks down for [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF]. Severe instability, however, remains a common feature to both problems.

Discrete Spurious Modes

The properties of the bilinear form s h (•, •) are crucial to the analysis of the discrete problem. We have seen that it is symmetric and semi-positive definite. Its kernel, denoted N (s h ), plays an important role in the study of [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF],

N (s h ) = ζ h ∈ H I,h : s h (ζ h , µ h ) = 0, ∀µ h ∈ H I,h . (20)
This space contains the "spurious modes" of s h (•, •). They prevent the bilinear form s h (•, •) from being positive definite. Indeed, when N (s h ) is reduced to {0}, the bilinear form s h (•, •) is positive definite on H I,h . As the dimension of H I,h is finite, this implies (non uniform) coerciveness of

s h (•, •) in H 1/2 (Γ I )
with a constant that depends on the mesh-size h and tends to zero with h, see Remark 5.4. It is instructive to compare with the continuous setting where the kernel N (s) = {0}, and s(•, •) does not suffer from spurious modes, although it is still not coercive due to the existence of "weak spurious modes".

This somewhat clarifies the issue of uniqueness. Most often, it is lost unless a discrete Holmgren principle is available. This principle and its link to the mesh will be discussed further on. Now, prior to existence, we have a first result proven in [START_REF] Azaïez | A Finite Element Model for the Data Completion Problem: Analysis and Assessment[END_REF].

Lemma 5.1 For all (g h , ϕ h ) in H C,h , the following orthogonality holds true

ζ h ∈ N (s h ) =⇒ ℓ h (ζ h ) = 0 .
Proof: Let us first prove the analogue of [START_REF] Ben Belgacem | Analysis of Lavrentiev-finite element methods for data completion problems[END_REF]. Indeed, considering that u N,h (µ h )u D,h (µ h ) vanishes on Γ I , the system (16) applied to ȗN,h (ϕ h ) can be tested with this function, thus giving,

Ω a∇ ȗN,h (ϕ h )•∇ u N,h (µ h )-u D,h (µ h ) dx = Γ C ϕ h u N,h (µ h )-u D,h (µ h ) d x = Γ C ϕ h u N,h (µ h ) d Γ since u D,h (µ h ) = 0 on Γ C
. Next, the system (16) applied to u N,h (µ h ) can also be tested with ȗD,h (g h ), as ȗD,h (g h ) = 0 on Γ I , and thus

Ω a∇ ȗD,h (g h ) • ∇ u N,h (µ h ) d x = 0,
because it corresponds to ϕ h = 0. Hence ℓ h (µ h ) can also be written for all µ h ∈ H I,h as

ℓ h (µ h ) = Ω a∇ ȗD,h (g h ) -ȗN,h (ϕ h ) • ∇ u N,h (µ h ) -u D,h (µ h ) d x. (21)
Now, let ζ h ∈ N (s h ). Then by choosing µ h = ζ h , applying [START_REF] Caubet | A dual approach to Kohn-Vogelius regularization applied to data completion problem[END_REF], and again the fact that u

N,h (ζ h ) - u D,h (ζ h ) = 0 on Γ I , we deduce that u N,h (ζ h ) = u D,h (ζ h ).
Owing to [START_REF] Clerc | Cortical mapping by Laplace-Cauchy transmission using a boundary element method[END_REF], this implies that ℓ h (ζ h ) = 0.

Remark 5.1 For another proof of Lemma 5.1, consider the energy functional associated to the discretized Kohn-Vogelius function (see Remark 2.1). We have the discrete analogues of ( 8) and ( 9),

J h (µ h ) = 1 2 s h (µ h , µ h ) -ℓ h (µ h ) = 1 2 |u D,h (µ h , g h ) -u N,h (µ h , ϕ h )| 2 a,H 1 (Ω) - 1 2 |ȗ D,h (g h ) -ȗN,h (ϕ h )| 2 a,H 1 (Ω) . (22)
The quadratic functional J h (•) has an infimum in the finite-dimensional space H I,h because it is convex and bounded below. The orthogonality statement of the Lemma follows directly from these observations. Details can be found in [ 17) is the Euler-Lagrange condition for the minimization of the energy functional J h (•) defined by [START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF], each of these solutions is also a minimizer of J h (•).

Our purpose is to find out, to the best of our ability, what happens when the mesh size tends to zero. Let us say it right away : we do not claim by any means that we will prove convergence of λ h to λ in the strong norm • s D . We believe that this is out of reach without additional regularization, and this will be commented later on. In this section, we focus on the finite elements spurious modes and their effect at the limit when h → 0. This is based on an important approximation result.

Beforehand, we state an elliptic regularity result satisfied by u N (µ). We refer to [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF] for elliptic regularity results in domains with corners.

Lemma 5.3 There exists a tubular neighborhood ω C of Γ C in Ω, away from Γ I , and a real number

q ∈]0, 1/2[ such that u N (µ) belongs to H 3/2+q (ω C ) and u N (µ) H 3/2+q (ω C ) ≤ C µ s D .
The constant C depends only on q.

Proof: The proof follows the same lines as in [START_REF] Ben Belgacem | Analysis of Lavrentiev-finite element methods for data completion problems[END_REF]Lemma 4.3]. The regularity exponent q is linked to the aperture of the largest inner angle at corners of the boundary Γ C .

Let

w(µ h ) = (u D -u N )(µ h ) and w h (µ h ) = (u D,h -u N,h )(µ h ).
The following lemma states a fundamental error estimate:

Lemma 5.4 There exists a real number p, 0 < p ≤ q, and a constant C independent of h such that

∀µ h ∈ H I,h , |w(µ h ) -w h (µ h )| a,H 1 (Ω) ≤ Ch 1/2+p µ h s D . ( 23
)
The number p depends on the inner angles of Γ.

Proof: The proof proceeds in three steps.

-First, observe that the function w(µ h ) solves

-div a∇ w(µ h ) = 0, in Ω, w(µ h ) = 0, on Γ I , w(µ h ) = -u N (µ h ) on Γ C . (24)
According to Lemma 5.3, the Dirichlet data, here (-u N (µ h )), belongs to H 1+q (Γ C ). By elliptic regularity, this implies that w(µ h ) ∈ H 3/2+p (Ω) for some real number p, 0 < p ≤ q, with

w(µ h ) H 3/2+p (Ω) ≤ C µ h s D . (25)
-Next, we estimate the approximation error of u N (µ h ) by u N,h (µ h ) on the boundary Γ C . We make use of Lemma 5.3 in the tubular neighborhood ω C away from Γ I so that the regularity is not limited by that of µ h . We have

u N (µ h ) -u N,h (µ h ) H 1/2 (Γ C ) ≤ Ch 1/2+q µ h s D . ( 26
)
The proof is pretty technical, and stems from the previous local estimates established by Nitsche and Schatz in [START_REF] Nitsche | Interior estimates for Ritz-Galerkin methods[END_REF]. We refer also to [START_REF]Full discretization of Cauchy's problem by Lavrentiev-finite element method[END_REF] for a detailed proof.

-Finally, w h (µ h ) ∈ X I,h is the standard finite element discretization of w(µ h ) solution to [START_REF] Feng | Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method[END_REF].

A standard numerical analysis yields (see [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF])

|w(µ h ) -w h (µ h )| a,H 1 (Ω) ≤ C( inf v h ∈X I,h |w(µ h ) -v h | a,H 1 (Ω) + u N (µ h ) -u N,h (µ h ) H 1/2 (Γ C ) ).
The final estimate [START_REF] Du | A Lavrentiev Finite Element Model for the Cauchy Problem of Data Completion: Analysis and Numerical Assessment[END_REF] follows by combining [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF], the regularity [START_REF] Friedman | Determining cracks by boundary measurements[END_REF], and standard results of finite elements approximation (see [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]). Now, we are in a position to quantify the difference in norms of the spurious modes.

Proposition 5.5 There exists a constant C independent of h such that

∀ζ h ∈ N (s h ), ζ h s ≤ C h 1/2+p ζ h s D . (27)
Proof: Let ζ h be given in H I,h . Owing to [START_REF] Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF], we have

ζ h 2 s = Ω a∇ u D (ζ h ) -u N (ζ h ) • ∇ u D (ζ h ) - Ω a∇ u D (ζ h ) -u N (ζ h ) • ∇ u N (ζ h ).
By construction of u N (ζ h ) and in view of (u

D (ζ h ) -u N (ζ h )) |Γ I = 0, the second integral vanishes
and leads to

ζ h 2 s = Ω a∇ u D (ζ h ) -u N (ζ h ) • ∇ u D (ζ h ).
The same argument applied to the full discretization to express ζ h s h yields a similar result. Then, we have that

ζ h 2 s h -ζ h 2 s = Ω a∇ u D,h (ζ h ) -u N,h (ζ h ) • ∇ u D,h (ζ h ) - Ω a∇ u D (ζ h ) -u N (ζ h ) • ∇ u D (ζ h ).
But

Ω a∇ u D (ζ h ) -u N (ζ h ) • ∇(u D (ζ h ) -u D,h (ζ h )) = 0, since u D (ζ h ) -u D,h (ζ h ) belongs to H 1 0 (Ω). Thus ζ h 2 s h -ζ h 2 s = Ω a∇ u D,h (ζ h ) -u N,h (ζ h ) • ∇ u D,h (ζ h ) - Ω a∇ u D (ζ h ) -u N (ζ h ) • ∇ u D,h (ζ h ) = Ω a∇ w h (ζ h ) -w(ζ h ) • ∇ u D,h (ζ h ).
This identity and the first part of (18) yield the bound

ζ h 2 s h -ζ h 2 s ≤ |w h (ζ h ) -w(ζ h )| a,H 1 (Ω) |u D,h (ζ h )| a,H 1 (Ω) ≤ C|w h (ζ h ) -w(ζ h )| a,H 1 (Ω) ζ h s D .
Considering Lemma 5.4, we derive that if they exist, act as weak spurious modes to the exact problem [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF]. Indeed, the normalization

ζ h 2 s h -ζ h 2 s ≤ Ch 1/2+p ζ h 2 s D . Now if ζ h ∈ N (s h ), it satisfies ζ h s h = 0,
ζ h := ζ h ζ h s D
produces weak normalized spurious modes, with the weak norm

ζ h s ≤ C √ h 1/2+p
that decreases to zero, with the mesh size h, while the strong norm is equal to unity,

ζ h s D = 1.
The proof of Proposition 5.5 may be adapted to show that the exact and discrete bilinear forms s(•, •) and s h (•, •) are close to each other.

Lemma 5.6 There exists a constant C independent of h such that sup

ζ h ∈H I,h sup µ h ∈H I,h s(ζ h , µ h ) -s h (ζ h , µ h ) ζ h s D µ h s D ≤ Ch 1/2+p . (28)
Proof: Let ζ h , µ h be given in H I,h . Arguing as in the proof of Proposition 5.5, we derive

s(ζ h , µ h ) -s h (ζ h , µ h ) = Ω a∇ w h (ζ h ) -w(ζ h ) • ∇ u D,h (µ h ).
Again [START_REF] Calvetti | GMRES, L-curves, and discrete ill-posed problems[END_REF] gives

s(ζ h , µ h ) -s h (ζ h , µ h ) ≤ |w h (ζ h ) -w(ζ h )| a,H 1 (Ω) |u D,h (µ h )| a,H 1 (Ω) ≤ C|w h (ζ h ) -w(ζ h )| a,H 1 (Ω) µ h s D .
And applying once again Lemma 5.4, we obtain

s(ζ h , µ h ) -s h (ζ h , µ h ) ≤ Ch 1/2+p ζ h s D µ h s D .
This proves (28).

Remark 5.3 It is possible to state a similar result for the linear forms ℓ(•) and ℓ h (•). Indeed, the following bound holds sup

µ h ∈H I,h ℓ(µ h ) -ℓ h (µ h ) µ h s D ≤ Ch 1/2+p .
The constant C depends on (g, ϕ), but not on h.

This error estimate and the one in Lemma 5.6 related to the finite element approximations of s(•, •)

and ℓ(•) are very important for the use and analysis of some regularizing methods. We refer to [START_REF]Full discretization of Cauchy's problem by Lavrentiev-finite element method[END_REF], for a comprehensive analysis of the combination of Lavrentiev procedure with finite element approximation.

Remark 5.4 Lemma 5.6 shows that s h cannot be uniformly elliptic in H 1/2 (Γ I ). Therefore, even if it had no spurious modes, it would still have weak spurious modes. Because the exact problem is ill-posed, all consistent numerical discretization of (6) will be plagued by weak spurious modes. In this respect, the situation is very different from the discretization of flows of incompressible fluids where some discretizations may give rise to weak spurious modes while others do not.

Discrete Holmgren Principle

Conditions that guarantee uniqueness of the solution to the finite element problem (17) are highly interesting. They will be derived from a discrete version of the Holmgren principle. To this end, we introduce the kernel

N h = v h ∈ X C,h , ∀w h ∈ X I,h , Ω a∇ v h • ∇ w h dx = 0 . ( 29
)
Definition 6.1 -The discrete Holmgren principle is said to be valid for problem [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF] if the kernel N h is reduced to zero.

Remark 6.1 The Holmgren principle, formulated in Definition 6.1, is straightforwardly related to the following variational form of problem (1) : find u ∈ H 1 (Ω) such that

Ω a∇ u • ∇ v dx = ϕ , v H 1/2 (Γ C ) ∀v ∈ H 1 I (Ω), u = g on Γ C .
This problem stems from the (variational) treatment of the data completion problem (1) (see [START_REF] Bourgeois | On mixed formulation of the Tikhonov regularization and its application to inverse PDE problems[END_REF]),

where the unknown and test functions are selected in different spaces. Indeed, in order to enforce the Neumann condition on Γ C , the test functions should not vanish there, while they must vanish on Γ I to compensate for the total lack of information on Γ I . Upon discretization, the stiffness matrix is rectangular, cannot be symmetric and has scarcely full rank. Therefore, existence is questionable, and can hardly be obtained at the discrete level.

As expected, the discrete Holmgren principle eliminates the spurious modes from the bilinear form s h (•, •) in equation ( 17).

Lemma 6.2 Assume that the discrete Holmgren principle holds true,

N h = {0}. (30)
Then, the kernel N (s h ) reduces to {0}. The bilinear form s h (•, •) has no spurious modes ; it is hence positive definite on H I,h .

Proof: Let ζ h ∈ N (s h ). According to [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF] and ( 19) we have,

s h (ζ h , ζ h ) = 0 = |u D,h (ζ h ) -u N,h (ζ h )| 2 a,H 1 (Ω) . Since (u D,h (ζ h ) -u N,h (ζ h )) vanishes on Γ I , this formula implies that u D,h (ζ h ) = u N,h (ζ h ).
In

Influence of the Mesh

The previous arguments are valid in two and three dimensions, but the present section is restricted to two dimensions, for simplicity. Let all finite element nodes of the mesh T h be numbered as

(x i ) 1≤i≤N .
They are the vertices of the triangles in T h . For the sake of convenience, let the numbering start with the nodes located on the complete boundary Γ C , i.e., (x i ) 1≤i≤n C , and the last ones be located on the incomplete boundary Γ I , i.e., (x i ) N -n I +1≤i≤N . The space X h is spanned by the usual hat shape functions (ψ i ) 1≤i≤N . The dimensions of the spaces are : dim

X h = N , dim H C,h = n C
, and dim H I,h = n I .

A Necessary Condition

The validity of the discrete Holmgren principle [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF] induces the following necessary condition on the mesh: Proof: First, we express (30) via a linear system. To this end, the degrees of freedom of the functions v h ∈ X C,h and w h ∈ X I,h are represented by vectors

v h = (v i = v h (x i )) n C +1≤i≤N ∈ R N -n C , w h = (w i = w h (x i )) 1≤i≤N -n I ∈ R N -n I .
With these notation, the discrete Holmgren principle (30) can be expressed in terms of the stiffness matrix of the discrete "Laplace problem" in ( 29)

(Av h = 0) =⇒ (v h = 0). ( 31
)
The coefficients of A are computed as the integrals

a i,j = Ω a∇ ψ j • ∇ ψ i dx, j ∈ n C + 1, N , i ∈ 1, N -n I .
Then, A has (Nn I ) rows and (Nn C ) columns. The assertion [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF] To achieve this, we need additional terminology, mainly borrowed from the basics of Graph Theory (see e.g. [START_REF] Balakrishnan | A Textbook of Graph Theory[END_REF]). The mesh T h is a simple non oriented triangular graph determined by the set

V h = (x i ) 1≤i≤N
of labeled vertices and the set of edges Ξ h . According to standard notation, we

can write T h = G(V h , Ξ h ).
For the sake of consistency, we will restrict the discussion to the "annular domain" in Fig. 1 -Two edges are adjacent if they share at least one common vertex.

-A simple line or a simple path is a sequence of p * vertices (x ip ) 1≤p≤p * , with no repeated vertices such that x i p-1 and x ip are adjacent. It may also be defined as the sequence of adjacent edges

[x i p-1 ,
x ip ] for all p (2 ≤ p ≤ p * ). Vertices (x i 1 , x ip * ) are the extreme points of the path and its length is (p * -1).

-A path is said to be closed or to be a cycle if [x ip * , x i 1 ] is also an edge. Each vertex is at the same time the tail and the head of the cycle. The length of the cycle is p * , the number of edges, equal to the number of vertices.

Now, we assume that we can construct a sequence of adjacent paths denoted by V

h , for k ≤ k * , that sweep the whole domain, according to the following process :

The initial path V (1) h = (x i ) i≤i C is determined by all nodes located on Γ C . By induction, V (k) h
is the sequence of vertices that do not belong to the previous paths

V (l) h (l < k), that are adjacent to at least one vertex in V (k-1) h . The symbol n k (= p * (k)) is used for the cardinal of the path V (k) h
or the number of vertices belonging to it.

The total number k * of the paths is finite and the last V (k * ) h contains the last nodes of Γ I . Note that some vertices of Γ I may be included in previous paths. However, we plan to describe the methodology and basic arguments rather than consider the most general meshes. Then, to avoid any unnecessary technical complications, we assume that the paths V (k) h are all closed, they are cycles, and we assume that V (k * ) h contains all nodes of Γ I . This configuration seems compatible with the domain plotted in Fig. 1 and is a reasonable working assumption.

These constructions settled, we revert to [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF]. Solving this equation is a step-by-step tracking of the frontal propagation of the zero information on the components v h of v h . The process starts with V . To simplify the discussion, let us extend by zero the vector v h at the vertices of Γ C and extend accordingly the stiffness matrix A of (31). The vector v h has k * k=1 n k = N components and the matrix A now has N columns, but it still has (Nn I ) rows because the test functions in the orthogonality relation in the definition of N h vanish on Γ I . A valuable knowledge can be gained from the bloc structure of A,

A =         A 1,1 A 1,2 (A 1,2 ) T A 2,2 A 2,3 (A k * -3,k * -2 ) T A k * -2,k * -2 A k-2,k * -1 (A k * -2,k * -1 ) T A k * -1,k * -1 A k * -1,k *         . (32)
The matrix A is rectangular and block tridiagonal. Diagonal blocks (A k,k ) k≤k * -1 are all symmetric positive definite, the super-diagonal ones (A k,k+1 ) k≤k * -1 contain the part of the full stiffness matrix modeling the interaction between the shape functions of the nodes in

V (k+1) h with those in V (k)
h . If the last column were dropped (forbidden for our target), the extracted matrix would be symmetric positive definite.

The necessary condition in Lemma 7.1 can be sharpened as follows: . This means

n I = n k * ≤ min k≤k * -1 n k .
Proof: The kernel N h is computed by equation [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF]. The vector v h is set into the same block structure as the block matrix A, that is

v h =      v (1)
v (2) . . .

v (k * )      ∈ R n 1 × R n 2 × • • • × R n k * .
In view of the block factorization A = LU in Lemma 11.3 (in the appendix), and given the invertibility of L, the linear system (31) can be transformed into U v h = 0, a rectangular "almost triangular" system

         S 1,1 v (1) + A 1,2 v (2) = 0 S 2,2 v (2) + A 2,3 v (3) = 0 . . . . . . S k * -1,k * -1 v (k * -1) + A k * -1,k * v (k * ) = 0
.

By taking into account the fact v (1) = 0 and applying the Gauss elimination, we arrive after (k * -1)

steps, at the final equation,

A 1,2 (S 2,2 ) -1 A 2,3 (S 3,3 ) -1 A 3,4 • • • (S k * -1,k * -1 ) -1 A k * -1,k * v (k * ) = 0. (33)
In order to conclude that necessarily v (k * ) = 0, we must show that the kernel of the matrix of this last equation is {0}. This is done by applying Lemma 11.4 (in the appendix) with

B k = A k,k+1 (S k+1,k+1 ) -1 for all k ≤ k * -2,

Spurious Modes and Lavrentiev Regularization

Here, we revert to the general dimension d = 2 or d = 3. The numerical simulation of ill-posed problems, in particular those severely ill posed, cannot be achieved safely without regularization.

The Lavrentiev technique proves to be highly effective for the variational problems ( 6) and ( 17), due to symmetry and non-negative definiteness of the operators (see [START_REF] Azaïez | On Cauchy's Problem. II. Completion, Regularization and Approximation[END_REF][START_REF] Azaïez | A Finite Element Model for the Data Completion Problem: Analysis and Assessment[END_REF][START_REF] Du | A Lavrentiev Finite Element Model for the Cauchy Problem of Data Completion: Analysis and Numerical Assessment[END_REF]). Let us clarify how the Lavrentiev Method handles the spurious modes, weak and strong, generators of instability, for both continuous and discrete settings.

Let ̺ be a small positive real number. Lavrentiev's method stabilizes (6) by adding a positive definite term as follows : find λ ̺ ∈ H 1/2 (Γ I ) such that,

̺s D (λ ̺ , µ) + s(λ ̺ , µ) = ℓ(µ), ∀µ ∈ H 1/2 (Γ I ). ( 34
)
The regularizing bilinear term ̺s D (•, •) is inserted to bring some strong coerciveness in the hope of dampening the weak spurious modes of the bilinear form s(•, •). Is λ ̺ infected by weak spurious modes? Predictably, the answer is no! At least not in a way to alter its quality. In fact, λ ̺ is, in some sense, orthogonal to these modes with respect to the s D (•, •)-inner product, as is shown by the next lemma.

Lemma 8.1 Let the sequence (ξ n ) n≥0 be a weak spurious mode of s(•, •), in the sense of Lemma 3.3.

Then, the following limit holds:

lim n→∞ s D (λ ̺ , ξ n ) = 0.
Proof: In view of [START_REF] Nitsche | Interior estimates for Ritz-Galerkin methods[END_REF], the sequence (ξ n ) n≥0 satisfies

∀n ≥ 0, ̺s D (λ ̺ , ξ n ) + s(λ ̺ , ξ n ) = ℓ(ξ n ).
By the definition of spurious modes, the term in s(•, •) tends to zero as n tends to infinity, and we have to examine the limit of (ℓ(ξ n )) n≥0 . The continuity of ℓ(•) with respect to the weak norm

• s (see [9, Lemma 2.1] ) implies that ℓ(ξ n ) ≤ C ξ n s ,
with C = C(g, ϕ). Hence (ℓ(ξ n )) n≥0 tends to zero as n tends to infinity. This proves the lemma. Now, the finite element problem [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF] is regularized as problem (34) : find λ ̺,h ∈ H I,h such that,

̺s D,h (λ ̺,h , µ h ) + s h (λ ̺,h , µ h ) = ℓ h (µ h ), ∀µ h ∈ H I,h . (35)
The next lemma shows that the Lavrentiev regularization filters the spurious modes of N (s h ). Its proof, which is skipped, is a trivial application of Lemma 5.1 and the definition (20) of N (s h ).

Lemma 8.2

The regularized solution λ ̺,h of problem [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF] belongs to the orthogonal subspace

N (s h ) ⊥ . That is s D,h (λ ̺,h , ξ h ) = 0, ∀ξ h ∈ N (s h ).
Remark 8.1 The regularized solution λ ̺,h is protected from the spurious modes of s h (•, •). The orthogonality keeps under control its strong norm, by preventing these modes from altering it. This explains the convergence of λ ̺,h towards the solution of ( 17) with the lowest s D (•, •)-norm (see [START_REF] Azaïez | A Finite Element Model for the Data Completion Problem: Analysis and Assessment[END_REF]).

Remark 8.2 We know from Proposition 5.5 that the possible spurious modes N (s h ) of s h (•, •)

behave like the weak spurious modes for s(•, •). It is almost certain that N (s h ) hides its own "weak spurious modes" that may be characterized by

ζ h ∈ N (s h ) ⊥ , lim h→0 ζ h s h ζ h s D,h = 0.
Their effect needs to be closely scrutinized in the light of regularization. As observed in [START_REF] Vandeven | Compatibilité des espaces discrets pour l'approximation spectrale du problème de Stokes périodique/non périodique[END_REF], in a different context -badly-conditioned discrete problems coming from well posed problems, then free of spurious modes, in the continuous setting-, these modes are more harmful than the spurious modes themselves. They are hard to identify, and then to cure.

Numerical Verification

In this section, we report some two-dimensional computations and draw significant observations that confirm the predictions made in the present work. The numerical simulations are carried out by means of a linear finite element program. A number of procedures come from the finite element library developed by O. Pironneau and presented in [START_REF] Lucquin | Introduction to scientific computing[END_REF]. The meshes are constructed using the EMC2 code (see [START_REF] Saltel | Un logiciel d'édition de maillage et de contours bidimensionnels[END_REF]).

The experiments are specifically focused on the erratic effect of the spurious modes on the computed solutions. The domain Ω is an annulus, delimited by two concentric circles centered at the origin and with radii (0.5, 1). The union of the inner and the outer circles defines the whole boundary. Each will be alternately the complete boundary Γ C while the other is the incomplete boundary Γ I . In all cases, the conduction coefficient a(•) equals one, i.e., a(•) = 1, and the exact solution is

u(x) = (x 1 + 1) 2 -(x 2 -1) 2 , ∀x ∈ Ω.
It is approximated by the discrete problem [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF]. After putting it in matrix form (S h λ h = ℓ h ), the resulting system is solved by the GMRES algorithm, strongly recommended for badly-conditioned problems (see [START_REF] Brown | GMRES on (nearly) singular systems[END_REF][START_REF] Calvetti | GMRES, L-curves, and discrete ill-posed problems[END_REF]).

The meshes built for the simulations are not of high resolution, they are however dense enough to reconstitute the annular geometry, as seen in Fig. 2. Notice that each mesh is based on three concentric circular cycles : one covering the outer circle, the other follows the inner one and an in-between cycle describing the middle circle.

∞ (u) = u -u h ∞ u ∞ ≈ 0.83 × 10 -2 .
The norm • ∞ stands for the maximum on the finite element vertices (x i ) 1≤i≤N . The results may be considered as plainly satisfactory, for this mesh, with Γ C having more nodes than Γ I .

(b) -The roles of the outer and inner circles are interchanged in the second experiment, regarding the boundary conditions. The inner line Γ C bears both Cauchy's conditions while the outer circle Γ I hosts the unknown, the missing condition. The size of problem ( 17) is 16 and the data are provided on a reduced number of vertices, namely 10 vertices. The rank of the stiffness matrix S h , is at most 10 by the Rank Theorem. In fact, according to Remark 7.1 the dimension of the spurious modes subspace is at least 6. GMRES is expected to be less stable than it is in the first test. The computations realized yield a relative error stalling at (e ∞ (u) ≈ 0.21). This level remains unchanged for any restarting parameter ≤ 10. As soon as it is larger, the error grows higher to (e ∞ (u) ≈ 0.59) (restart parameter = 11), then to (e ∞ (u) ≈ 1.27) (GMRES restart = 12), before blowing up. A possible explanation of this behavior is that since the rank of the stiffness matrix S h is 10, the dimension of the Krylov subspace is expected to be also 10. Moreover, in view of the existence result for [START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF], the right-hand side should be in the range R(S h ). Theoretically, beyond ten iterations, the Krylov subspace fully covers the range subspace R(S h ), the orthogonalization procedure, within GMRES, produces the zero mode, and the algorithm should stop. But, floatingpoint arithmetics possibly produces non-vanishing modes orthogonal to the Krylov subspace, and hence to the range R(S h ). Because of symmetry, those modes are in the kernel N (S h )(= N (s h )), the subspace of the spurious modes. If GMRES is not restarted, the Krylov subspace will be augmented by some of those spurious modes and the minimization fails to give a reliable solution. The conclusion is that instability is produced by the fact that there are more nodes on Γ I than on Γ C .

(c) -The third test uses the mesh, to the right of Fig. 2. The outer line is Γ I and the inner one is Γ C . The cycles V

(1) h and V

(3) h of the mesh, seen as a graph, contain n 1 = n 3 = 16 vertices each, their common length. The length of the middle cycle is set to 10. The size of the stiffness matrix is 16. But its rank does not exceed ten. We realized several runs, proceeding as in the second test. The relative error is (e ∞ (u) ≈ 0.32) for a restarting parameter ≤ 10. It is (e ∞ (u) ≈ 0.64) for restarting after 11 iterations, and (e ∞ (u) ≈ 1.21) for a restarting parameter equal to 12, then grows up indefinitely. Hence, instability may be caused by the internal structure of the mesh.

These trends are all in agreement with the theory. As soon as spurious modes are activated, the result is instability generation and the computations are inaccurate. The quality of the discrete solution is altered to an important extent. Regularization becomes mandatory to stabilize the simulations. A possible remedy, though not the best, is to limit the effect of the spurious modes by interrupting the iterative algorithm at an early stage. Sometimes, the results enjoy some accuracy.

Lavrentiev Computations

In this paragraph, we focus on the way the Lavrentiev method filters the spurious modes and examine its positive effect on the computed solution. We only consider the last two experiments that suffer from instability. The regularization parameter is almost zero, ̺ = 10 -13 , but it is sufficient to ensure the orthogonality to the spurious modes, of the computed missing Dirichlet data λ h,̺ , solution to [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF].

(b ′ ) -We begin by revisiting experiment (b), where the stabilizing Lavrentiev term is added to problem [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF]. The numerical results show a substantial improvement on the discrete solution. Indeed, the relative error drops to (e ∞ (u) ≈ 0.03). This illustrates the tremendous repercussions of regularization in getting rid of the severe effect of the spurious modes. Notice that the computations remain stable even if the algorithm is stopped after hundreds of iterations.

(c ′ ) -After the Lavrentiev regularization is applied to the third example (c), the relative error drops to (e ∞ (u) ≈ 0.05), The GMRES algorithm is interrupted after thousands of iterations, and the restart parameter is set to 15. As expected, the performance of the Lavrentiev method is highly satisfactory.

These simple examples highlight the decisive action of the regularization, produced by the Lavrentiev method here, in purging the discrete solution from the harmful effect of the spurious modes. Together with similar procedures they are mandatory tools for safe computations of severely ill-posed problem such as the Cauchy-Poisson equation addressed in this paper.

Conclusion

In this work, we have focused on the discrete version of the Holmgren uniqueness principle, for the finite element approximation of the Cauchy-Poisson inverse problem. The finite element solution exists for arbitrary Cauchy boundary data, while uniqueness is not guaranteed, and even fails very often. The continuous and the finite element settings offer then two contrasted configurations. The pivotal influence of the triangular meshes is investigated in this work, as we show that uniqueness may be restored for some particular meshes. We provide valuable insights on what happens and formalize some necessary conditions on that class of performing meshes for ensuring that the discrete problem has exactly one solution. This reduces the causes of instability, when it comes to solve the discrete problem, and shows the strength and relevance of the regularizing procedures advocated here.

To our belief this is the starting step, our quest is still on the road for more extensions, in several directions (space dimension, finite element degree, . . . ), and possibly, and most significantly, to come up with a sufficient condition that possibly determines a broad class of suitable meshes for the discrete version of Holmgren theorem to hold. Before ending, let us mention that, after a deep bibliography research, we are not aware that these issues have ever been addressed before.

This is a symmetric and positive definite matrix. Each of its diagonal square blocks (A k,k ) k<k * , with size (n k ) k<k * , inherits these same properties.

In the sequel, we need a collection of the Schur complement matrices, defined by the induction formula : for all k ≤ k * -1,

S 1 = A 1,1 , S k = A k,k -(A k-1,k ) T (S k-1 ) -1 A k-1,k , k ≥ 2.
Then, the decomposition of the matrix à can be written by means of both the diagonal and the lower triangular matrices Assume first that the construction of the Schur complement matrices is possible, and in particular that they are all invertible. The proof that à = LSL T can be done by a computation of blockstructured matrix products. As a direct consequence, the matrices (S k ) 1<k * are all positive definite. This is verified step-by-step during the factorization, because these (S k ) are the pivot matrices of the (k * -2) iterations of the Gauss elimination. The procedure starts by observing that the first block pivot S 1,1 = A 1,1 . As A 1,1 is positive definite, the first step can be realized. Then, we assert that S 2,2 , the pivot matrix of the second step, inherits definiteness from that of à ; it is thus invertible and the second elimination can proceed. This step is general and hence the factorization proceeds rigorously by induction.

Let Ũ be the upper triangular matrix Ũ = SL T , and let U k denote the block row number k of the matrix Ũ

Ũ =       S 1,1 A 1,2 S 2,2 A 2,3 A k * -2,k * -1 S k * -1,k * -1       =       U 1 U 2 U k * -2 U k * -1       .
Then the LU factorization of à is à = L Ũ .

To conclude, we introduce the following rectangular upper triangular matrix

U = U [k * -2] U k * -1 A k * -1,k * .

Remark 5 . 2

 52 and (27) follows from the above formula. Proposition 5.5 implies that the spurious modes (ζ h ) h≥0 of the discrete problem[START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF],

Lemma 7 . 1

 71 If the discrete Holmgren principle (30) is true, then n C ≥ n I .

7. 2

 2 Meshes and Graphs. More Necessities Lemma 7.1 is a good starting point for acquiring more information on the role of the mesh on the validation (or violation) of the discrete Holmgren principle. To further this knowledge, we investigate how the mesh propagates the zero values of v h (nullity information), originally prescribed at the nodes of the complete boundary Γ C . That propagation is of course driven by the orthogonality in[START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF].

  , in particular both borders Γ C and Γ I are closed lines. As far as the meshes are concerned, the extension to adjacent boundaries Γ C and Γ I is straightforward. Let us provide more simplifying assumptions to identify further necessary conditions while avoiding technical complications. Definition 7.2 (see [6, Chapter 1]) -Two vertices joined by an edge are said to be adjacent. They are both vertices of at least one triangle. Two vertices are neighbors if they are adjacent.

  of Γ C ) and advances steadfastly, cycle-by-cycle, towards V (k * ) h (the nodes of Γ I ). The k-th iteration transmits the nullity information (on v h ) from the cycle V

Lemma 7 . 3

 73 Assume the discrete Holmgren principle (30) holds true. Then, any cycle in the sequence (V (k) h ) k≤k * has a higher length than the last one V (k * ) h

Remark 7 . 2 Remark 7 . 3

 7273 and B k * -1 = A k * -1,k * and it gives n k ≥ n k * = n I , for all k ≥ 1. The proof is complete. Note that the length of intermediate cycles must be compared to n k * = n I , the length of the last cycle V (k * ) h. Comparing mutually the lengths of intermediate cycles brings no useful information. Indeed, in the nullity progress from one cycle to the next, the quantity of information that is gained or lost does not matter, provided that each cycle inherits sufficient information to reach the desired conclusion for the cycle on Γ I , and by then ensures the finite element Holmgren principle. The methodology exposed here extends as well to three dimensions, at the cost of a little more complexity. The main modifications is that paths are replaced by open surfaces and cycles by closed surfaces.

Figure 2 : Meshes 9 . 1

 291 Figure 2: Meshes

S 1 (A 1 , 2 ).

 112 = Diag[(S k,k ) 1≤k≤k * -1 ], T (S 1 ) -1 I 2 (A k * -2,k * -1 ) T (S k * -2 ) -1 I k * -1Clearly, the matrix L is invertible and its determinant is one. Then we have the following result: Lemma 11.2 The matrix à the factorization à = LSL T .Proof: It is based on Gauss elimination. We only sketch the argument, without going into details.

  Remark 6.2 A by product of the proof is :ζ h is a spurious mode of s h (•, •), i.e., ζ h ∈ N (s h ), if it is the trace on Γ I of a "volumetric spurious mode" v h ∈ N h , in the sense that ζ h = v h|Γ I . The orthogonality in N h expresses two properties. One is that u D,h (ζThe positive-definiteness of s h (•, •) derived in Lemma 6.2 yields the invertibility of the variational finite element problem[START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF]. Under the discrete Holmgren principle[START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF], the discrete problem (17) has exactly one solution λ h ∈ H I,h .

	Remark 6.3 Proposition 6.3

turn, this implies that u N,h (ζ h ) vanishes on Γ C , hence belongs to X C,h . By comparing with

[START_REF] Han | The finite element method in a family of improperly posed problems[END_REF]

, we see that it belongs to N h ; then by assumption

[START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF]

, u N,h (ζ h ) = 0. This implies that ζ h = 0.

The proof is complete. h ) is harmonic in the variational finite element sense. The other is that it satisfies the homogeneous Neumann condition on Γ C , in the discrete sense. The homogeneous Dirichlet condition on Γ C is imposed on u D,h (ζ h ) by construction as it belongs to X C,h .

  Remark 7.1 A consequence of lemma 7.1 is : for problem[START_REF] Burman | Primal-dual mixed finite element methods for the elliptic Cauchy problem[END_REF] to have a unique solution, it is necessary that the number of data (dim H C,h ) be larger than the number of unknowns (dim H I,h ).Notice that if the condition n C ≥ n I is violated, the dimension of the kernel N h (and of the subspace of spurious modes N (s h )) is at least (n In C ).

states that the matrix A has a trivial kernel ; hence, it is injective. By the Rank-Nullity Theorem, this occurs only if A has more rows than columns, i.e., (Nn I ) ≥ (Nn C ), which means that n I ≤ n C .

This means that both n and m go to ∞.

Appendix : Some Technical Results

This appendix is devoted to three mathematical tools. The proofs of two of them (Lemmas 11.1 and 11.4) can be found in the literature, but not in the form expressed here. Hence, their proofs are sketched.

"Weak" Cauchy's Convergence Test

Let H be a Hilbert space. A sequence (z n ) n≥0 ⊂ H is said to fulfill the Cauchy convergence criterion with respect to the weak topology of

We propose to establish that this sequence has a weak limit in H. This result is needed in the proof of Lemma 3.3. Such a result is known in a more general Banach setting. It may be established using various theorems such as the Banach-Alaoǧlu theorem or the Banach-Steinhauss theorem.

The reader will find below a short proof in a Hilbert setting.

Lemma 11.1 Assume that the sequence (z n ) n≥0 ⊂ H satisfies the Cauchy convergence criterion with respect to the weak topology. Then it has a weak limit z in H.

Proof: For any y ∈ H, (z n , y) H is a Cauchy sequence in R. Therefore it converges to a limit, say l(y) ∈ R. It is easy to check that l(•) is a linear functional and its continuity is a direct consequence of the uniform boundedness of the Banach-Steinhauss theorem. By the Riesz representation theorem we find that l(y) = (z, y) H for some z ∈ H. Finally, we obtain that

And z is then the weak limit of the sequence (z n ) n≥0 ⊂ H. The proof is complete.

Factorization of the Rectangular Block Structured A

Let us establish the factorization of the rectangular block structure matrix A (given in [START_REF] Koshev | FEM-based scalp-to-cortex EEG data mapping via the solution of the Cauchy problem[END_REF]) and required in the proof of Lemma 7. Its size is Nn I and its tridiagonal block structure comes from the block structure of A,

The symbol U [k * -2] is used for the matrix of the (k * -2) first rows of Ũ . The matrix U has the same size as A. It has (k * -1) block rows and k * block columns. Note that the upper triangularity here is linked to the upper triangular structure of the matrix Ũ , expanded over the block rows of the first block column. We have then the following decomposition Lemma 11.3 The matrix A has the factorization A = LU .

Proof: The decomposition can be readily checked by using the LU decomposition of à and by observing that

The proof is complete.

Kernel of Product of Matrices

The investigation on the discrete Holmgren principle used the following matrix result in the proof of Lemma 7.3. The proof is based on the Rank-Nullity Theorem, also called Rank Thoerem.

Lemma 11.4 Let m * ≥ 2 and (B m ) 1≤m≤m * be a sequence of matrices of R nm,n m+1 for 1 ≤ m ≤ m * .

The following kernel properties hold: The proof is complete.