A Appendix

Functional A Posteriori Error Estimates for Elliptic Problems in Exterior Domains

Abstract

This paper is concerned with the derivation of computable and guaranteed upper bounds of the difference between the exact and the approximate solution of an exterior domain boundary value problem for a linear elliptic equation. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for problems in bounded domains of (see [2, 3]).
Key Words A posteriori error estimates of functional type, elliptic boundary value problems in exterior domains
AMS MSC-Classifications 65 N 15

1 Introduction

The main focus of our investigations is to suggest a method of deriving guaranteed and computable upper bounds of the difference between the exact solution of an elliptic exterior domain boundary value problem and any approximation from the corresponding energy space. We discuss the method with the paradigm of the prototypical elliptic problem

(1.1)
(1.2)

We assume that with is an exterior domain, i.e. is compact, with Lipschitz continuous boundary (see Figure 1).

Figure 1: exterior domain with artificial interface

Throughout this paper we will use the weighted Lebesgue function spaces

Here and denotes the radius vector. is a Hilbert space equipped with the scalar product

where and belong to and is Lebesgue’s measure. We denote the corresponding norms by . If then coincides with the usual Lebesgue space . For the sake of simplicity we keep the same notation for spaces of vector-valued functions. Moreover, we introduce the weighted Sobolev space

which is a Hilbert space as well with respect to the scalar product

By we denote the closure of , the space of compactly supported smooth test functions, in the norm of . Whenever we consider Sobolev spaces of bounded domains we use the usual unweighted -scalar products and -norms.

For dimensions the solution theory for the problem (1.1)-(1.2) is based on the weighted Poincare/Friedrich estimate (see Corollary 16 (i) and Remark 17 of the appendix)

(1.3)

the Lax-Milgram theorem and, if needed, an adequate extension operator for the boundary data. Let be some function in satisfying the boundary condition (1.2). The weak solution of (1.1)-(1.2) is then defined by the variational formulation

(1.4)

By (1.3) the left hand side of (1.4) is a strongly coercitive sesqui-linear form over provided that the real-matrix-valued function is measurable, bounded a.e., symmetric and uniformly strongly elliptic, i.e.

(1.5)

If then by the Cauchy-Scharz inequality the right hand side of (1.4) is a linear and continuous functional over . Thus, under these assumptions the problem (1.4) is uniquely solvable in by Lax-Milgram’s theorem.

If one can apply the same arguments with the difference that (1.3) has to be modified. For and, for example, we have by Corollary 16 (iii) and Remark 17

(1.6)

Hence, we get the same solution theory with tiny restrictions on , which easily can be removed by a translation. For the singularities are stronger and additionally we have to utilize logarithmic terms. By Corollary 16 (ii) and Remark 17 we have for domains , such that the complement contains the unit ball,

(1.7)

where

is a Hilbert space equipped with the natural scalar product

and again denotes the closure of in the norm of . Consequently, we obtain for all with and all in satisfying the boundary condition (1.2) a unique solution belonging to .

We summarize the results in the following

Theorem 1

Let as well as and satisfying the boundary condition (1.2). Then the exterior boundary value problem (1.1)-(1.2) is uniquely weakly solvable in . The solution operator is continuous.

From the above discussion, it is clear that for the existence of weak solutions in suitable spaces can also be proved.

Remark 2

The boundary data and its extension can be described in more detail. In the bounded domain case it is well known that there exists a bounded linear trace operator and a corresponding bounded linear extension operator (right inverse) mapping to and vice verse. Hence, by restriction we get a bounded linear trace operator

and by extension and applying an obvious cutting technique we obtain a bounded linear extension operator

for our exterior domain , which even maps to functions with (arbitrarily thin) compact support. As in the bounded domain case, is a right inverse of . Then we may specify and as well as our variational formulation for : Find , such that

Finally, we introduce

which is a Hilbert space with respect to the canonical scalar product

2 Upper bounds for the deviation from the exact solution in dimensions

Let be an approximation of , where is assumed just to belong to since the boundary condition may not be satisfied exactly. Our goal is to obtain upper bounds for the difference between and in terms of the norm

We use (1.4) and get for all

(2.1)

Before we proceed we note two useful results.

Theorem 3

Let be as above. Moreover, let be a linear and continuous functional over and , such that for all

holds. Then

(2.2)
for all , for which coincides with on the boundary . If additionally belongs to then
(2.3)

Proof We consider

Using Cauchy-Schwarz’ inequality we obtain

and thus . By the triangle inequality we get (2.2). (2.3) is trivial since we can set , i.e. .

We may be more specific using the trace and extension operators from Remark 2.

Corollary 4

Let the assumptions of Theorem 3 be satisfied. Then

Here is the constant in the inequality

(2.4)

Proof Setting and as well as using (2.4) proves the inequalities. We note that (2.3) follows directly from the corollary as well.

In the subsequent sections we introduce and discuss some different functionals and corresponding constants .

2.1 First estimate

For any and any we have

(2.5)

Combining (2.1) and (2.5) we obtain for all and all

(2.6)

By Cauchy-Schwarz’ inequality, (1.3) with and (1.5) we estimate the right hand side of (2.6) as follows:

(2.7)
(2.8)

By Corollary 4 we arrive at the following result.

Proposition 5

Let be as in Theorem 3. Then

(2.9)

where is an arbitrary vector field in .

Remark 6

If satisfies the prescribed boundary condition, then (2.9) implies

(2.10)

The estimates (2.9) and (2.10)) show that deviations from exact solutions of exterior boundary value problems have the same structure as for problems in bounded domains, namely they contain weighted residuals of basic relations with weights given by constants in the corresponding embedding inequalities.

2.2 Second estimate

Assume that is decomposed into two subdomains and with interface (see Figure 1) and that the fields exactly satisfy the relation

(2.11)

In particular, this situation may arise if the source term has compact support and is represented (in the exterior domain ) as a linear combination of solenoidal fields having proper decay at infinity. In this case, the estimate of Proposition 5 turns trivially to

(2.12)

which holds for all additionally satisfying (2.11), where the weight constant is

(2.13)

which follows directly from

But we also may derive another estimate. We rewrite (2.7) and use Cauchy-Schwarz’ inequality in

(2.14)

and estimate

(2.15)

Here denotes a Poincare/Friedrich constant associated with the bounded domain , i.e. the best constant of the inequality

where denotes the trace operator. In this case, we have again (2.12) but now with the (optional) weight constant

(2.16)

We note that the constant (2.13) may also be achieved by (2.7) and the argument (2.14) if we replace the estimate (2.15) by

We summarize and get our second a posteriori error estimate.

Proposition 7

For all with (2.11) we have

where is defined either by (2.13) or by (2.16).

Remark 8

In general, the number will be smaller and thus provides a better bound than . On the other hand, the number is an easily computable upper bound for the best possible constant .

2.3 Third estimate

Let and be the restrictions of some to and , respectively. Assuming and but not necessarily we use the equations

(2.17)
(2.18)

which hold for all and in the sense of the traces and respectively . At this point we assume that the interface is Lipschitz (in order to guarantee that the traces are well defined). By we denote the duality product of and . We recall that the normal traces and possess weak surface divergences in as well. If , then in and in . Hence, in this case adding (2.17) and (2.18) we obtain by (2.5)

for all . Therefore, we get

for all since is surjective.

On our way to find like in (2.6) we now insert (2.17), (2.18) instead of (2.5) into (2.1) and obtain

(2.19)

The third term of will be estimated by (2.8) and for the last term we may use the continuity of the trace operator in combination with a Poincare/Friedrich estimate, i.e.

(2.20)

and obtain

(2.21)

To estimate the second term of we again use (1.3) and (1.5) and obtain

(2.22)

Considering the first (and last) term of we have once more at least two options as in section 2.2 to obtain the estimate

(2.23)

with defined either by (2.13) or (2.16).

Finally with (2.19) and (2.8), (2.21), (2.22), (2.23) we get by Corollary 4 the third estimate.

Proposition 9

For all with and we have

(2.24)

with from Proposition 7. The right hand side of (2.24) vanishes if and only if coincides with and with .

Remark 10

There are many ways to deduce (2.20). We just mention that can be considered as a trace of a function defined in or or even of a function, which is just defined in a small neighborhood of . Thus, we may adjust the constant according to our needs.

Remark 11

This estimate suggests even a solution method: We construct approximations using locally supported trial functions in , e.g. FEM, and utilize global approximations properly behaving at infinity for . These two types of approximations are usually difficult to meet together exactly on the artificial boundary . However, Proposition 9 shows that this is not required because we can use instead the penalty term with known penalty factor . In addition, we have one more parameter, the ‘radius’ of the interface . Since is artificial and arbitrary we can use this parameter in the algorithm in order to obtain better results.

Remark 12

At this point we shall note that all our estimates are sharp, which easily can be seen by setting and .

Remark 13

In Propositions 5, 7, 9 we can always replace the last summand of the right hand side by or using Theorem 3 and Corollary 4.

3 Upper bounds in dimension

Of course, Theorem 3 holds for as well and the modifications on the estimates depend just on the Poincare/Friedrich estimate and thus they are obvious using the proper Cauchy-Schwarz inequality. We achieve

Proposition 14

Let , such that contains the unit ball.

  • For all

  • For all with in

    where .

  • For all with and

Analogously, Remarks 8, 10, 11, 12, 13 hold.

Appendix A Appendix

a.1 Lower bounds for the error

We note by a standard variational argument

Thus, we obtain for all the estimate

which is sharp since one can put . But to exclude the unknown exact solution from the right hand side we need since then by (1.4)

(A.1)

But this estimate is no longer sharp because we can not put anymore. In fact, with and we get for

Hence, we obtain the estimate

for all , which is sharp and coincides with (A.1) if . But the unknown exact solution still appears on the right-hand side, i.e. the normal trace of on . Furthermore, if then (A.1) can not be sharp.

a.2 Poincare type estimates for exterior domains

We introduce the radial derivative , where . Furthermore, and denote the open ball and sphere of radius centered at the origin in , respectively. We will use the ideas of [4, Lemma 4.1] and [1, Poincare’s estimate III, p. 57] with some minor useful modifications.

Lemma 15

Let , , be a domain and . For all the following Poincare estimates hold:

  • If then

  • Let . If or then

  • If then

    where will be extended by zero to .

For the estimates derived in this paper it suffices to set . In this particular case, the above lemma implies

Corollary 16

Let , , be a domain. For all the following Poincare estimates hold:

  • If then

  • If and then

  • If then

    Hence, if we have

Remark 17

Of course, by continuity all these estimates extend to appropriate weighted -Sobolev spaces.

Proof Let , , be a domain and . By partial integration we get for all and

Thus, for all and

Now the left hand side of this equality converges by the monotone convergence theorem. Since , if and only if , and the right hand side converges for by Lebesgue’s dominated convergence theorem in . Hence, for we obtain

Choosing we finally get by the triangle inequality

Since we are especially interested in the case this estimate is only applicable in dimensions .

For we proceed as follows: For all we have