On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

06/20/2019 ∙ by Laurent Bourgeois, et al. ∙ Inria ENSTA ParisTech 0

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε>0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

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1 Introduction and setting of the problem

Figure 1: Examples of domains . The thick blue lines represent the support of measurements.

Let us consider a bounded Lipschitz domain , , the boundary of which is partitioned into two sets and . More precisely, and are non empty open sets for the topology induced on from the topology on , and (see Figure 1). The Cauchy problem we are interested in consists, for some data , in finding such that

(1)

where is the outward unit normal to . This kind of problem arises when some part of the boundary of a structure is not accessible, while the complementary part is the support of measurements which provide the Cauchy data . It is important to note that in practice those measurements are contaminated by some noise. Due to Holmgren’s theorem, the Cauchy problem (1) has at most one solution. However it is ill-posed in the sense of Hadamard: existence may not hold for some data , as for example shown in [3]. A possibility to regularize problem (1) is to use the quasi-reversibility method, which goes back to [31] and was revisited in [26]. The original idea was to replace an ill-posed Boundary Value Problem such as (1) by a family, depending on a small parameter , of well-posed fourth-order BVPs. Much later, the first author introduced the notion of mixed formulation of quasi-reversibility for the Cauchy problem of the Laplace equation [4]. This notion was extended to general abstract linear ill-posed problems in [7]. The idea is to replace the ill-posed second-order BVP by a family, again depending on a small parameter , of second-order systems of two coupled BVPs: the advantage is that the order of the regularized problem is the same as the original one, which is interesting when it comes to the numerical resolution. The price to pay is the introduction of a second unknown function in addition to the principal unknown . Such mixed formulation of quasi-reversibility is the following: for , find such that for all ,

(2)

where , and . In (2), the brackets stand for duality pairing between and . Here is the subspace formed by the functions in which, once extended by on , remain in . We observe that in view of Poincaré inequality, the standard norm of in the spaces and is equivalent to the semi-norm defined by . Let us denote the corresponding scalar product. We remark that the weak formulation (2) is equivalent to the strong problem

(3)

where we observe that the two unknowns and are harmonic functions which are coupled by the boundary . We have the following theorem.

Theorem 1.1.

For all , the problem (2) has a unique solution . There exists a constant which depends only on the geometry such that

If in addition we assume that is such that problem (1) has a (unique) solution (the data are said to be compatible), then there exists a constant which depends only on the geometry such that

and

To prove such theorem, we need the following Lemma, which establishes an equivalent weak formulation to problem (1) and which is proved in [7].

Lemma 1.1.

For , the function is a solution to problem (1) if and only if and for all with , we have

(4)
Proof of Theorem 1.1.

Let us begin with the first part of the theorem. There exists a continuous lifting operator from to such that . Let us define . By replacing in (2), we obtain that satisfies, for all , the system

Well-posedness then relies on the Lax-Milgram Lemma applied to the coercive bilinear form

on . Choosing and and subtracting the two above equations, we obtain

The Cauchy-Schwarz inequality implies

The equivalence of norm and the standard norm in spaces and , the continuity of the trace operator and the continuity of the lifting operator yield

Using the Young’s inequality to deal with the right hand side of the above inequality, the result follows. Let us prove the second part of the theorem. In the case when the Cauchy data is associated with the solution , then satisfies the weak formulation (4). By subtracting (4) to the second equation of (2), we obtain that for all ,

(5)

Now setting in the first equation of (2), setting in equation (5) and subtracting the two obtained equations, we get

We deduce that the term in the above sum is nonpositive, which from the Cauchy-Schwarz inequality implies that and then . Hence there exists a constant such that

It remains to prove that in when . The sequence is bounded in . Therefore, there exists a subsequence, still denoted , such that in when , with . Since the affine space is convex and closed, it is weakly closed. This guarantees that . Besides, by passing to the limit in the second equation of (2) we obtain that satisfies the weak formulation (4). Uniqueness in problem (1) then implies that , so that weakly converges to in . But

so that weak convergence implies strong convergence. Lastly, a standard contradiction argument enables us to conclude that all the sequence strongly converges to in . ∎

Remark 1.1.

Let us mention that another type of mixed formulation of quasi-reversibility was introduced in [20], in which the additional unknown lies in instead of . In addition, a notion of iterative formulation of quasi-reversibility was introduced and analyzed in [19]. We believe that the quasi-reversibility formulation (2) is the easiest one to handle to establish regularity results of the weak solutions.

The estimates of Theorem 1.1 involve norms of the regularized solution in the case of a Lipschitz domain and for the natural regularity of the Cauchy data , that is . These estimates were derived in two different cases: the data are compatible or not. The main concern of this paper is to analyze, when the domain and the Cauchy data are more regular than Lipschitz and , respectively, the additional regularity of the solution , whether the data are compatible or not. We also want to obtain estimates in the corresponding norms. In order to simplify the analysis, the additional regularity of the data is formulated in the following way: we assume that is such that there exists a function in with and that we can define a continuous lifting operator . Denoting and considering the new translated unknown , the initial Cauchy problem (1) can be transformed into a homogeneous one (however still ill-posed): for , find such that

(6)

We emphasize that this regularity assumption made on the data is not an assumption of regularity of the solution . It is simple to construct smooth data in the sense above such that the corresponding is only in and not in . The mixed formulation of quasi-reversibility for problem (6) takes the following form: for , find such that for all ,

(7)

Note that the strong equations corresponding to problem (7) are

(8)

The analog of Theorem 1.1, the proof of which is skipped, is the following.

Theorem 1.2.

For all and , the problem (7) has a unique solution . There exists a constant which depends only on the geometry such that

(9)

If in addition we assume that is such that problem (6) has a (unique) solution , then there exists a constant which depends only on the geometry such that

(10)

and

The objective is now to study the regularity of the solution to problem (7) and to complete the statements (9) and (10) of Theorem 1.2 by giving estimates in stronger norms. One objective, as will be seen in section 6, is the following. In practice, one has to solve problem (7) in the presence of two approximations. Firstly, the data is altered by some noise of amplitude . Secondly, the problem (7) is discretized, for instance with the help of a Finite Element Method based on a mesh of size . It is then desirable to estimate the error between the approximated solution and the exact solution as a function of , and . Such error estimate for the norm needs the solution to be in a Sobolev space , with . It could be noted that in a recent contribution [13] (see also [9, 10, 11, 12]), a discretized method was proposed in order to regularize the Cauchy problem (1) in the presence of noisy data without introducing a regularized problem such as (7) at the continuous level. In some sense, the method of [13] relies on a single asymptotic parameter, that is , instead of two in our method, that is and . However, we believe that from the theoretical point of view, the regularity of quasi-reversibility solutions is an interesting problem in itself. To our best knowledge, it has never been investigated up to now. The difficulty stems from the fact that we analyze the regularity of a problem involving a small parameter which degenerates when tends to . There are other contributions (see e.g. [25, 18, 15, 16, 35, 36]) where regularity results or asymptotic expansions are obtained in situations where the limit problem has a different nature from the regularized one. For example in [18], the authors study a mixed Neumann-Robin problem where the small parameter is the inverse of the Robin coefficient. But while both the perturbed problem and the limit one are well-posed in [18], only the perturbed problem is well-posed in our case, the limit problem being ill-posed (in any framework). Our contribution is original in this sense. In the present work, we study the regularity of the solution of the regularized problem as tends to zero. We emphasize that computing an asymptotic expansion of the solution with respect to and proving error estimates (for example as in [24, 32]) remains an open problem, the reason being that, due to the ill-posedness of the limit problem, no result of stability can be easily established.

Our paper is organized as follows. First we consider the simple case of a smooth domain in Section 2, where classical regularity results (see for example [8]) can be used. The case of the polygonal domain is introduced in Section 3, where we also analyze the regularity of the quasi-reversibility solution in corners delimited by two edges of or two edges of . In this case, the regularity of functions and can be analyzed separately with the help of the classical regularity results of [22] in a polygon for the Laplace equation with Dirichlet or Neumann boundary conditions. In Section 5 we consider the more difficult case of a corner of mixed type, that is delimited by one edge of and one edge of . This analysis relies on the Kondratiev approach [27], which is based on some properties of weighted Sobolev spaces which are recalled in Section 4. Section 6 is dedicated to the application of our regularity results to derive some error estimate between the exact solution and the quasi-reversibility solution in the presence of two perturbations: noisy data and discretization with the help of a Finite Element Method. Two appendices containing technical results, which are used in Section 5, complete the paper. The main results of this article are Theorem 2.1 (uniform regularity estimates in smooth domains), Theorem 3.1 (uniform regularity estimates in 2D polygonal domains) and the final approximation analysis of Section 6.

2 The case of a smooth domain

Let us first assume that is a domain of class . If , then there exists a function such that and even a continuous lifting operator from to (see Theorem 1.5.1.2 in [21]). We are therefore in the situation described in the Section 1, where the problem to solve is (6). We begin with an interior regularity result.

Proposition 2.1.

For , the solution to the problem (7) is such that for all , and belong to and there exists a constant which depends only on the geometry such that

If in addition is such that problem (6) has a solution , then

where the norm is defined by

Proof.

From the first equation of (8), we have that

Clearly

, which by using the Fourier transform implies that

and hence

From (9) we obtain that

If in addition is such that problem (6) has a (unique) solution , from (10) we obtain

The estimates of are obtained following the same lines. ∎

Let us now establish a global regularity estimate (up to the boundary) in the restricted case when (see Figure 1 right).

Theorem 2.1.

For , the solution to the problem (7) is such that and belong to and there exists a constant which depends only on the geometry such that

If in addition is such that problem (6) has a solution , then

Proof.

Given , we may find two infinitely smooth functions and such that in a vicinity of and in a vicinity of . We have from the first equation of (8),

Since on , from a standard regularity result for the Poisson equation with Dirichlet boundary condition we obtain

(11)

and from (9) we have

From a standard continuity result for the normal derivative and using on , we obtain

From the second equation of (8) we have

Combining the two previous estimates with the fact that on implies the regularity estimate

Reusing the second equation of (8), the estimate (9) and that on leads to

and using on , we obtain

We conclude that

Now let us assume that is such that problem (6) has a solution . From (10) and (11) we now have the better estimate

Using on , we obtain

and then

Reusing the second equation of (8), the estimate (10) and that on leads to

Since on , we obtain

We conclude that . ∎

Remark 2.1.

From Theorem 1.1 and Proposition 2.1, we notice that in the interior of the domain, the estimates are the same as the estimates, whether the data are compatible or not. However, from Theorem 1.1 and Theorem 2.1, when it comes to the estimates in the whole domain, up to the boundary, one loses a factor with respect to the estimates, whether the data are compatible or not.

3 The case of a polygonal domain

3.1 Main result

Figure 2: An example of polygonal domain. , , represent the three types of vertices that we will study in §3.2, §5, §3.3 respectively.

From now on, is a polygonal domain in dimension 2. Our motivation is indeed to obtain error estimates in the context of the discretization with the help of a classical Finite Element Method: due to the meshing procedure in two dimensions, in practice the computational domain is often a polygon. We use the same notations as in [22] to describe the geometry of such a polygon. Let us assume that is the union of segments , , where is an integer. Let us denote the vertex such that , the angle between and from the interior of , the unit tangent oriented in the counter-clockwise sense and the outward normal to . We assume that and are formed by a finite number of edges, namely and , respectively, with . Let us denote the subset of functions such that , , with the following compatibility conditions at :

(12)

and the equivalence at means that for small

where denotes the point of which, for small enough (say ), is at distance (counted algebraically) of along . More precisely, if and if . It is proved in [22], that for , there exists a function such that for each , and even a continuous lifting from to . We are hence again in the framework of section 1, where the problem to solve is (6).

Clearly, the interior estimates given by Proposition 2.1 are true in the polygonal domain since they are independent of the regularity of the boundary. Let us now analyze the regularity up to the boundary. As done in [22], the estimates are obtained by using a partition of unity, which enables us to localize our analysis in three different types of corners (see Figure 2):

  • regularity at a corner delimited by two edges which belong to , called a corner of type ,

  • regularity at a corner delimited by two edges which belong to , called a corner of type ,

  • regularity at a corner delimited by one edge which belongs to and one edge which belongs to , called a corner of mixed type.

Let us denote by the set of such that is either a vertex of type or a vertex of type and the set of such that is a corner of mixed type. We wish to prove the following theorem, which is obtained by gathering Propositions 2.1, 3.1, 3.2 and 5.1 hereafter.

Theorem 3.1.

Let us take if there exists such that and otherwise. Let us take if there exists such that and otherwise. Let us denote .

For and , the solution to the problem (7) is such that and belong to and there exists a constant which depends only on the geometry such that

If in addition we assume that is such that problem (6) has a (unique) solution , then

Remark 3.1.

The global estimates of Theorem 3.1 are obtained by gathering all the local estimates obtained in Propositions 2.1, 3.1, 3.2 and 5.1. Each of these estimates are locally better than the global estimate of Theorem 3.1.

3.2 Regularity at a corner of type

The regularity of solutions and near a corner delimited by two edges which belong to can be analyzed separately. They will be obtained by directly applying the results of [22] for Dirichlet and Neumann Laplacian problems. Let us consider the vertex of a corner delimited by two edges and which belong to . Let us denote the local polar coordinates with respect to the point and a radial function (depending only on ) such that for and for . We assume that is chosen such that in a vicinity of all edges except for or . In order to simplify notations, we skip the reference to index , denoting in particular , and . Let us introduce the finite cone . The two following lemmata are proved in [22].

Lemma 3.1.

For , the problem: find such that

(13)

has a unique solution and there exists a unique constant and a unique function such that

Moreover, there exists a constant such that

In addition, if then .

Lemma 3.2.

For , the problem: find such that