A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems

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1 Computational Methods in Applied Mathematics Vol ), No. 3, pp c 013 Institute of Mathematics, NAS of Belarus Doi: /cmam A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems Willy Dörfler Stefan Sauter Abstract We develop a new analysis for residual-type a posteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber > 0 as our model problem. We employ a classical conforming Galerin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., ), pp ], Melen and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only O d ) degrees of freedom, where d = 1,, 3 denotes the spatial dimension. In the present paper, we will introduce an a posteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber > 0 provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the a posteriori estimates would be amplified by a factor. 010 Mathematical subject classification: 35J05, 65N1, 65N30. Keywords: Helmholtz Equation at High Wavenumber, Stability, Convergence, hp-finite Elements, A Posteriori Error Estimation. 1. Introduction In this paper, we will introduce a new analysis for residual-based a posteriori error estimation. We consider the conforming Galerin method with hp-finite elements applied to a class of highly indefinite boundary value problems, which arise, e.g., when electromagnetic or acoustic scattering problems are modelled in the frequency domain. As our model problem we consider a highly indefinite Helmholtz equation with oscillatory solutions. Residual-based a posteriori error estimates for elliptic problems have been introduced in [4, 5] and their theory for elliptic problems is now fairly completely established cf. [1, 19]). To setch the principal idea and to explain the goal of this paper let u denote the unnown) solution of the wea formulation of an elliptic second order PDE with appropriate boundary conditions. Typically the solution belongs to some infinite-dimensional Sobolev space H. Let u S denote a computed Galerin solution based on a finite dimensional subspace S H. Willy Dörfler Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology KIT), 7618 Karlsruhe, Germany willy.doerfler@it.edu. Stefan Sauter Institut für Mathemati, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland stas@math.uzh.ch.

2 334 Willy Dörfler, Stefan Sauter A reliable) a posteriori error estimator is a computable expression η which depends on u S and the given data such that an estimate of the form u u S H Cηu S ) 1.1) holds for a constant C which either is nown explicitly or sharp upper bounds are available. We emphasize that in the literature various refinements of this concept of a posteriori error estimation exist while for the purpose of our introduction this simple definition is sufficient. In the classical theory the constant C depends linearly on the norm of the solution operator of the PDE in some appropriate function spaces, more precisely, it depends reciprocally on the inf-sup constant γ cf. [5, Theorem 3.]). In [1] it was proved for the Helmholtz problem with Robin boundary conditions that for certain classes of physical domains the reciprocal inf-sup constant 1/γ and, hence, also the constant C in 1.1)) grows linearly with the wavenumber. See also [9] for further estimates of the inf-sup constant for the Helmholtz problem. However, this implies that for large wavenumbers the classical a posteriori estimation becomes useless because the error then typically is highly overestimated. Additional difficulties arise for the a posteriori error estimation for highly indefinite problems because the existence and uniqueness of the classical Galerin solution is ensured only if the mesh width is sufficiently small. In contrast to definite elliptic problems, there exist only relatively few publications in the literature on a posteriori estimation for highly indefinite problems cf. [, 3, 10, 18]). In [14] and [15] a new a priori convergence theory for Galerin discretizations of highly indefinite boundary value problems has been developed which is based on new regularity estimates the splitting lemmas as in [14] and [15]), where the solution u = u rough + u osc is split into a rough part and a smooth part with high-order regularity in weighted) Sobolev spaces. Under appropriate assumptions on the smoothness of the data domain Ω, right-hand side f) the following regularity estimates hold: u rough H Ω) C f and u osc H m Ω) C f m 1, where the constant C f is independent of. This theory allows in the a priori convergence theory to absorb the -depending L -error of the PDE into the wavenumber-independent part of the equation. As a consequence, discrete stability and pollution-free a priori convergence estimates can be proved under a very mild resolution condition on the hp-finite element space which requires only O d ) degrees of freedom. In this paper, we will develop a new a posteriori analysis based on similar ideas: The L - part of the a posteriori error will be estimated by the H 1 -error and then can be compensated by an appropriate choice of the hp-finite element space. This allows to prove reliability and efficiency of the error estimator. The constants in these estimates become independent of the, possibly, high wavenumber > 0 provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the a posteriori estimates would be amplified by a factor. The paper is structured as follows. In Section, we will consider as our model problem the high frequency, time harmonic scattering of an acoustic wave at some bounded domain in an unbounded exterior domain and transform it to a finite domain by using a Dirichlet-to- Neumann boundary operator or some approximation to it. We define a conforming Galerin hp-finite element discretization for its numerical approximation. In Section 3, we summarize the a priori analysis as in [14] and [15] which will be needed to determine the minimal hp-finite element space for a stable Galerin discretization.

3 A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems 335 In Section 4, we will present the a posteriori error estimator and prove its reliability and efficiency. It will turn out that the optimal polynomial degree p will depend logarithmically on the wavenumber and, hence, the finite element interpolation theory has to be explicit with respect to the mesh width h and the polynomial degree p. In a forthcoming paper, we will focus on numerical experiments and also on the definition of an hp-refinement strategy in order to obtain a convergent adaptive algorithm.. Model Helmholtz Problems and their Discretization.1. Model Problems The Helmholtz equation describes wave phenomena in the frequency domain which, e.g., arises if electromagnetic or acoustic waves are scattered from or emitted by bounded physical objects. In this light, the computational domain for such wave problems, typically, is the unbounded complement of a bounded domain Ω in R d, d = 1,, 3, i.e., Ω out := R d \Ω in. Throughout this paper, we assume that Ω in has a Lipschitz boundary Γ in := Ω in. The Helmholtz problem depends on the wavenumber. In most parts of the paper exceptions: Remars 3.5, 4. and Corollaries 4.10, 4.11) we allow for variable wavenumber : Ω out R but always assume that is real-valued, nonnegative, and a positive constant outside a sufficiently large ball cf..7)). For a given right-hand side f L Ω out ), the Helmholtz problem is to see U Hloc 1 Ωout ) such that )U = f in Ω out.1a) is satisfied. Towards infinity, Sommerfeld s radiation condition is imposed: r U i U = o x 1 d ) for x,.1b) where r denotes differentiation in radial direction and the Euclidian vector norm. For simplicity we restrict here to the homogeneous Dirichlet boundary condition on Γ in : U Γ in = 0..1c) We assume that f is local in the sense that there exists some bounded, simply connected Lipschitz domain 1 Ω such that a) Ω in Ω, b) suppf) Ω, and c) is constant in a neighborhood of Ω and anywhere outside Ω. The computational domain cf. Figure 1) will be Ω := Ω \Ω in with boundary Γ in Γ out and Γ out := Ω. It is well nown that problem.1) can be reformulated as: Find u H 1 Ω) such that )u = f in Ω, u = 0 on Γ in, n u = T u on Γ out,.) 1 Since Ω in is bounded, Ω can always be chosen as a ball. Other choices of Ω might be preferable in certain situations.

4 336 Willy Dörfler, Stefan Sauter 9 / K J / E 9 E 9 K J 9 Figure 1. Scatterer Ω in with boundary Γ in and exterior domain Ω out. The support of f is assumed to be contained in the bounded region Ω. The domain for the wea variational formulation is Ω = Ω \Ω in. where T denotes the Dirichlet-to-Neumann operator for the Helmholtz problem and n is the normal derivative at Γ out. The previous problems are posed in the wea formulation given by: Find u H := {u H 1 Ω) u Γ in = 0}.3) such that A DtN u, v) := Ω u, v u v) T u) v = Γ out Ω fv for all v H..4) Since the numerical realization of the nonlocal DtN map T is costly, various approaches exist in the literature to approximate this operator by a local operator. The most simple one is the use of Robin boundary conditions leading to )u = f in Ω, u = 0 on Γ in, n u = i u on Γ out..5) The wea formulation of this equation is given by: Find u H such that A Robin u, v) := u, v u v) i u v = fv for all v H..6) Ω Γ out Ω In most parts of this paper, we allow indeed that is a function varying in Ω. In scaling diamω in ) = 1, the wavenumber is dimensionless. The following conditions are always assumed to be satisfied: L R d, R), 0 essinf x Ω x) esssup x) =: max <, x Ω = const > 1 outside a large ball, = const in a neighborhood U const of Γ out and anywhere outside Ω..7) For given data g H 1/ Γ out ), the function u g H 1 loc Rd \Ω ) is well defined as the wea solution of )u = 0 in R d \Ω which satisfies Sommerfeld s radiation conditions as well as u g = g on Γ out. Then T g := n u g.

5 A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems 337 We expect that the condition > 1 in.7) can be weaened to 0 for our a posteriori error estimates as long as the Dirichlet part Γ in has positive measure. However, the low frequency case 0 1 corresponds to the proper elliptic case and can be treated by using the well established theory as, e.g., developed in [16]. Let U const := U const Ω. The constants in the estimates in this paper will depend on max, and U const through a trace inequality as in Lemma.6) but hold uniformly for all functions satisfying.7)... Abstract Variational Formulation Notation.1. All functions in this paper are in general complex-valued. For a Lebesgue measurable set ω R d and p [1, ], m N, we denote by L p ω) the usual Lebesgue space with norm L p ω), scalar product, ) L ω) for p =, and by H m ω) the usual Sobolev spaces with norm H m ω). The seminorm which contains only the derivatives of highest order is denoted by H m ω). We equip the space H, defined in.3), with the norm v H;Ω := v L Ω) + +v L Ω) )1/ with + := max{1, } which is obviously equivalent to the H 1 Ω)-norm. Both sesquilinear forms A DtN.4) and A Robin.6) belong to the following class of forms see Proposition.5). Assumption. Variational formulation). Let Ω R d, for d {, 3}, be a bounded Lipschitz domain. Then H, equipped with the norm H;Ω, is a closed subspace of H 1 Ω). We consider a sesquilinear form A : H H C that can be decomposed into where av, w) := Ω A = a b, v, w v w) = v, w) L Ω) v, w) L Ω) and the sesquilinear form b satisfies the following properties: a) b : H H C is a continuous sesquilinear form with for some positive constant C b. bv, w) C b v H;Ω w H;Ω for all v, w H, b) There exist -independent numbers θ 0 and γ ell > 0 such that the following Gårding inequality holds: ReAv, v)) + θ + v L Ω) γ ell v H;Ω for all v H..8) c) The adjoint problem: Given g L Ω), find z H such that Av, z) = v, g) L Ω) for all v H.9) is uniquely solvable and defines a bounded solution operator Q : L Ω) H, g z, more precisely, the -dependent) constant is finite. C adj Q := sup +g) H;Ω g L Ω)\{0} + g L Ω).10)

6 338 Willy Dörfler, Stefan Sauter Remar.3. Note that the different scaling of the constant C adj with respect to compared adj to the corresponding constant, say C, in [14, 4.4)] and [15, 3.10)] becomes necessary since we allow for non-constant wavenumbers here and cannot tae as a simple multiplicative scaling factor out of the norms in.10). For constant, the relation of these constants is adj = C. C adj Problem.4. Let A be a sesquilinear form as in Assumption.. For given f L Ω) we see u H such that Au, v) = f, v) L Ω) for all v H..11) Proposition.5. Both sesquilinear forms A Robin.6) and A DtN.4) under the additional condition that Γ out is a sufficiently large sphere) satisfy Assumption. with γ ell = 1 and θ =. Proof. The proof is a slight modification of the corresponding proofs for constant wavenumber in [14] and [1]. Condition a) for A Robin will follow from Lemma.6. For A DtN we employ that is constant in U const and Γ out is a sphere of a large radius R > 0. Hence, from the proof of [14, Lemma 3.3] it follows that T u, v) Γ out C R 1 u H 1/ Γ out ) v H 1/ Γ out ) + const u L Γ out ) v L Γ out )). By again using Lemma.6 we obtain T u, v) Γ out C ) C R tr u H;Ω v H;Ω and the continuity of A DtN follows. For condition b) and Robin boundary conditions, we employ ReA Robin v, v)) + + v L Ω) Ω v + + v + + ) v ) v H;Ω and.8) holds with θ = and γ ell = 1. For the sesquilinear form A DtN, we employ [14, Lemma 3.3 )] to obtain ReA DtN v, v)) + + v L Ω) v Ω + + v + + ) v ) ) ReT u, v) Γ out) v H;Ω and.8) again holds with θ = and γ ell = 1. For condition c), we may apply Fredholm s theory and, hence, it suffices to prove that au, v) bu, v) = 0 for all v H.1) implies u = 0. For Robin boundary conditions, we argue as in [1, 8.1.)] and for DtN boundary conditions as in the proof of [14, Theorem 3.8] to see that.1) implies u H0Ω). 1 Hence, u solves u, v u v) = 0 for all v H. Ω

7 A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems 339 Let Ω be a bounded domain such that Ω Ω R d \Ω in and Γ out Ω. The extension of u by zero to Ω is denoted by u 0. It satisfies u HΩ ) := {u H 1 Ω ) u Γ in = 0} and u 0, v u 0 v) = 0 for all v HΩ ). Ω Elliptic regularity theory implies that u 0 H Q) for any compact subset Q Ω, in particular, in an open Ω neighborhood of Γ out. The unique continuation principle cf. [11, Chapter 4.3]) implies that u 0 = 0 in Ω so that u = 0 in Ω. Lemma.6. There exists a constant C tr depending only on U const such that v H 1/ Γ out ) C tr v H;Uconst and v L Γ out ) C tr v 1/ L U const) v 1/ H 1 U const) For as in.7) we have v L Γ out ) C tr v H;Uconst C tr v H;Ω for all v H 1 Ω) for all v H 1 Ω). for all v H 1 Ω). Proof. The first two inequalities are standard Sobolev trace estimates [8, Lemma 3.1]. To prove the third inequality we note that due to = const on U const, there holds.3. Discretization const u L Γ out ) C tr const u L U const) u H 1 U const) C tr const u L U + const) u H 1 U const)) 1 + const ) u L U + const) u H 1 U const)) = C tr Ctr + u L U + const) u H 1 U const)) Conforming Galerin Discretization. A conforming Galerin discretization of Problem.4 is based on the definition of a finite dimensional subspace S H and is given by: Find u S S such that Au S, v) = f, v) L Ω) for all v S..13).3.. hp-finite Elements. As an example for S as above, we will define hp-finite elements on a finite element mesh T consisting of simplices K T which are affine images of a reference simplex. More precisely, T is a partition of Ω into d-dimensional disjoint open simplices which satisfy Ω = K T K and, for any two non-identical elements K, K T, the intersection K K either is empty, a common vertex interval endpoint in 1-d), a common edge for d ), or a common face for d = 3). The local mesh width is given by the diameter h K of the simplex K and p K denotes the local polynomial degree on K. Assumption.7. Concerning the local mesh widths h K and local polynomial degrees p K, it will be convenient cf. [16, 10)]) to assume that they are comparable to those of neighboring elements: There exist constants c shape, c deg > 0 such that c 1 shape h K h K c shape h K, c 1 deg p K p K c deg p K for all K, K T with K K =..14)

8 340 Willy Dörfler, Stefan Sauter The first requirement is also referred to as shape-regularity and is a measure for possible distortions of the elements. Many of the constants in the following estimates will depend on c shape, c deg. Definition.8 hp-finite element space). For meshes T with element maps F K as in Assumption.7 the hp-finite element space of piecewise mapped) polynomials is given by S p T ) := { v H : v K F K P pk for all K T }, where P q denotes the space of polynomials of degree q N. For chosen T and polynomial degree vector p, we may let S = S p T ). 3. A Priori Analysis In this section, we collect those results on existence, uniqueness, stability, and regularity for the Helmholtz problem.4), which later will be used for the analysis of the a posteriori error estimator Well-Posedness Proposition 3.1. Let Ω in R d, d =, 3, in.1a) be a bounded Lipschitz domain which is star-shaped with respect to the origin. Let Γ out := B R for some R > 0. Then.4) admits a unique solution u H for all f H which depends continuously on the data. Proposition 3.. Let Ω be a bounded Lipschitz domain. For all f H 1 Ω)), a unique solution u of problem.6) exists and depends continuously on the data. For the proofs of these propositions for constant we refer, e.g., to [1, Proposition 8.1.3] and [7, Lemma 3.3], while for variable one may argue as in Proposition Discrete Stability and Convergence An essential role for the stability and convergence of the Galerin discretization is played by the adjoint approximability which has been introduced in [15]; see also [6, 17]. Note that the scaling with respect to the wavenumber differs from the scaling in the quoted papers for the same reasons as for Q as explained in Remar.3.) Definition 3.3 Adjoint approximability). For a finite dimensional subspace S H, we define the adjoint approximability of Problem.4 by σ S) := sup g L Ω)\{0} inf v S Q +g) v H;Ω, 3.1) + g L Ω) where Q is as in.10). Theorem 3.4 Stability and convergence). Let γ ell, θ, C b, C adj S as in Section.3.1. Then the condition σs) γ ell θ1 + C b ) be as in Assumption. and 3.) implies the following statements:

9 A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems 341 a) The discrete inf-sup condition is satisfied: inf sup v S\{0} w S\{0} Av, w) v H;Ω w H;Ω γ ell + γ ell /1 + C b ) + θc adj b) The Galerin method based on S is quasi-optimal, i.e., for every u H there exists a unique u S S with Au u S, v) = 0 for all v S, and there holds u u S H;Ω γ ell 1 + C b ) inf v S u v H;Ω, + u u S ) L Ω) γ ell 1 + C b ) σ S) inf v S u v H;Ω. Proof. The proof follows very closely the proofs of [14, Theorems 4., 4.3] considering variable. As an example we show the intermediate result of + u u S ) L Ω) 1 + C b ) u u S H;Ω σ S). For e = u u S and z = Q +e) it holds that + e L Ω) = +e, e) L Ω) = Ae, z) = Ae, z z S ) > C b ) e H;Ω z z S H;Ω 1 + C b ) e H;Ω σ S) + e L Ω). Remar 3.5. In [14, 15], it is proved for constant wavenumber, that for S as in Section.3., i.e., hp-finite elements, the conditions p = Olog)) and h p = O1) 3.3) imply 3.), i.e., σ S) Proof of [14, Corollary 5.6] and [15, Proposition 5.3] C 1 + h p ) h h ) p ) 3.3) p + σp γ ell θ1 + C b ), where the constants in the O ) terms in 3.3) depend only on the -independent constants γ ell, θ, C b, and σ. This leads to the finite element space with only O d ) degrees of freedom for a stable discretization of the Helmholtz equations. In this light, terms in the a posteriori error estimates which grow polynomially in p are expected to grow, at most, logarithmically with respect to and, hence, are moderately bounded, also for large wavenumbers. 4. A Posteriori Error Estimation The following assumption collects the requirements for the a posteriori error estimation. Assumption 4.1. a) The continuous Helmholtz problem satisfies Assumption.. b) S is a hp-finite element space as explained in Section.3. and satisfies Assumption.7. c) u S S is the computed solution satisfying the Galerin equation.

10 34 Willy Dörfler, Stefan Sauter Remar 4.. a) Assumption 4.1 does not require the stability condition 3.) to be satisfied which is only sufficient for existence and uniqueness of the discrete problem. We only assume that u S exists, is computed, and solves the Galerin equation for the specific problem. To be on the safe side in the case of constant wavenumber, one can start the discretization process with the a priori choice 3.3) of p and h which implies 3.) and, in turn, the existence and uniqueness of a Galerin solution for any right-hand side in L Ω). b) The constant in the a posteriori error estimate will contain the term σ S) as a factor. In order to get an explicit upper bound, an a priori estimate of the quantity is required which can be found for constant wavenumber in [14, Theorem 5.5] and [15, Propositions 5.3, 5.6] cf. Remar 3.5). For a simplicial finite element mesh T, the boundary of any element K T consists of d 1)-dimensional simplices. We call the relatively open interior of) these lower dimensional simplices the edges of K, although this terminology is related to the case d =. The set of all edges of all elements in T is denoted by E. The subset E E consists of all edges which are contained in Γ out while the subset E Ω E consists of all edges that are contained in Ω. Finally, we set E := E Ω E, the set of all edges that are not in Γ in. For a K T we define simplex neighborhoods about K by ω 0) K ω j) K := {K}, := { K K T and K ω j 1) K } for j 1, E K := {E E E K}. 4.1) Definition 4.3 Residual). For v S we define the volume residual resv) L Ω) and the edge residual Resv) L E E E) by resv) := f + v + v on K T, { [ n v] E on E E Ω, Resv) := n v + i v on E E. Here [v] E is the jump of the given function v on the edge E, i.e., the difference of the limits in points x E from both sides. In the definitions above we used exact data f,. We will later, Section 4., replace these by approximations. The residual Resv) is defined for the Robin boundary condition.5) for simplicity. With an obvious modification of this definition, we could also insert a term T v here, instead of i v), for the DtN boundary condition.). Definition 4.4 Error estimator). We define for v S the error indicator ηv) := αk resv) L K) + 1/, αe Resv) L E)) 4.) E E K T where the weights {α K, α E K T, E E} are given by α K := h K p K, α E := he p E ) 1/.

11 A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems 343 The choice of the weights α K, α E in 4.) is related to an interpolation estimate in [16, Theorems.1,.] which we explain next. Theorem 4.5. Let Ω R and let p = p K ) K T denote a polynomial degree distribution satisfying.14). Let Assumption 4.1 b) be satisfied. Then there exists a constant C 0 > 0, that depends only on the shape-regularity of the mesh cf. Assumption.7), and a bounded linear operator I S : Hloc 1 R ) S such that for all simplices K T, all edges E E K, and v H we have p K pe v I S v L h K) + v I S v L K h E) C 0 v L ω 4) E K ). 4.3) Proof. This result has been proven in [16] in a vertex oriented setting, but is easily reformulated as stated above using shape uniformity and quasi-uniformity in the polynomial degree. Theorem 4.8 will show that ηu S ) can be used for a posteriori error estimation. That it estimates the error from above is called reliability, that it estimates the error from below is called efficiency Reliability According to Assumption 4.1 the exact solution u H and the Galerin solution u S S of.11) and.13), respectively, exist. In view of inequality.8), we estimate the error e = u u S, ReAe, e)), and + e L Ω) separately in terms of ηu S ). Lemma 4.6. Let Assumption 4.1 be satisfied. Then there is a constant C 1 > 0, that depends only on the shape-regularity of the mesh cf. Assumption.7), such that ReAe, e)) C 1 ηu S ) e H;Ω. Proof. We start with an auxiliary computation. Let e = u u S denote the Galerin error and let w H. By using solution properties, e.g., the Galerin orthogonality and integration by parts, we obtain the error representation Ae, w) = Ae, w I S w) = K T resu S ), w I S w) L K) + E EResu S ), w I S w) L E). We use the assumed interpolation estimates 4.3) and get with the Cauchy Schwarz inequality, applied to integrals and sums, Ae, w) C 0 αk resu S ) L K) + ) 1/ αe Resu S ) L E) E E K T K T w H;ω 4) K C 1 ηu S ) w H;Ω. 4.4) By setting w = e and using ReAe, w)) Ae, w) we obtain the assertion. Lemma 4.7. Let Assumption 4.1 be satisfied. Then, with C 1 from Lemma 4.6 and σ S) as in 3.1), + e L Ω) C 1 σ S)ηu S ). 4.5) ) 1/

12 344 Willy Dörfler, Stefan Sauter Proof. We define z by.9) with g := +e. Let z S S denote the best approximation of z with respect to the H;Ω -norm. By using 4.4) with w = z z S and the definition of the adjoint approximation property, we obtain and this gives 4.5). + e L Ω) = Ae, z) = Ae, z z S) C 1 ηu S ) z z S H;Ω C 1 ηu S )σ S) + g L Ω) Theorem 4.8 Reliability estimate). Let Assumption 4.1 be satisfied. Then, with C 1 from Lemma 4.6, e H;Ω 1 γ ell C γell θ) 1/ σ S) ) ηu S ). Proof. The combination of.8), 4.5) with the bounds obtained in Lemmata 4.6 and 4.7 yields so that γ ell e H;Ω ReAe, e)) + θ + e L Ω) C 1ηu S ) e H;Ω + θc 1σ S) ηu S ) e H;Ω 1 θ ) 1/ C 1 ηu S ) + C1 σ γ ell γ S)ηu S ) ell 1 C γell θ) 1/ σ γ S) ) ηu S ). ell In the previous arguments res and Res were defined with exact data functions f,. In order to obtain two-sided bounds for the Galerin error in terms of an error estimator one has to modify the error estimator by data oscillations: For any K T we define the local and global data oscillations by osc K := α K f f L ω K ) + ) )u S L ω K ) and osc := osc K. The modified error estimator ηu S ) is defined by replacing f and in the definitions of res and Res in 4.3) by local L K)-projections f, of degree p K or some degree q K p K. Corollary 4.9. Let Assumption 4.1 be satisfied. Then the modified error estimate is reliable up to data oscillations: 3 e H;Ω C γell θ) 1/ σ γ S) ) ηu S ) + osc ). ell Proof. We notice that and, on Γ out, K T resu S ) = f + u S + u S = f + u S + u S + f f + )u S = resu S ) + f f + )u S Resu S ) = n u S + i u S = Resu S ), since is constant on Γ out. The same argument applies to T u S. We thus obtain ηu S ) 3 ηu S ) + 3 K T α K f f L K) + 3 K T α K )u S L K). Hence, the assertion follows from Theorem 4.8.

13 A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems 345 An explicit estimate of the error by the error estimator requires an upper bound for the adjoint approximation property σ S). Such estimates for hp-finite elements spaces for constant wavenumbers are derived in [14] and [15] for problem.4) and.6), respectively. We summarize the results as the following corollaries. Corollary 4.10 Robin boundary conditions). Consider problem.6) with constant wavenumber, where Ω R d, d {, 3}, is a bounded Lipschitz domain. Either Ω has an analytic boundary or it is a convex polygon in R with vertices A j, j = 1,..., J. We use the approximation space S described in Section.3.. If Ω is a polygon, then the hp-finite element space S is employed where, in addition, L = Op) geometric mesh grading steps are performed towards the vertices for the details we refer to [15]. Let f L Ω) and > 1 and assume that Γ in =, i.e., we consider the pure Robin problem. Let Assumption 4.1 a) and b) be satisfied. Then there exist constants δ, c > 0 that are independent of h, p, and such that the conditions h δ and p 1 + c log) p imply 3.1) and thus the -independent a posteriori error estimate where Č only depends on δ and c. e H;Ω C Č ) ηu S ), Corollary 4.11 DtN boundary conditions). Consider problem.4) for constant wavenumber, where Ω has an analytic boundary. Let Assumption 4.1 a) and b) be satisfied and assume that the constant C adj in.10) grows at most polynomially in, i.e., there exists some β 0 such that 3 C adj C β. Let f L Ω) and > 1. Then there exist constants δ, c > 0 that are independent of h, p, and such that the conditions h δ and p 1 + c log) p imply 3.1) and thus the -independent a posteriori error estimate e H;Ω C Č ) ηu S ), where Č only depends on δ and c. 4.. Efficiency The reliability estimate in the form of Theorem 4.8 allows us to use the error estimator as a stopping criterion in order to satisfy a certain given accuracy requirement. In order to achieve efficiently this goal one has to ensure that the true error is not overestimated too much by the error estimator and that its local spatial distribution is well represented. This is called efficiency. It is verified by estimating the local error by the localized error estimator. In this light, we define the localized version of the error estimator by η K v) := αk resv) L K) + 1 ) 1/, α E Resv) L E) E E K with E K as in 4.1). Note that ηv) = K T η K v). 3 See [9] for sufficient conditions on the domain which implies this growth condition.

14 346 Willy Dörfler, Stefan Sauter As in Corollary 4.9 let us define approximations f, to f,, respectively, as local L K)- projections onto a polynomial of degree p K or some q K p K ). In this case, we use the notation res and η accordingly. Also we set K,+ := max{ L K), 1} Theorem 4.1. Let Assumption 4.1 and.7) be satisfied and let the mesh be shape regular cf. Assumption.7). We assume that Ω is either an interval d = 1), or a polygonal domain d = ), or a Lipschitz polyhedron d = 3), and that the element maps F K are affine. We assume the resolution condition: K,+ h K p K 1 for all K T. 4.6) Then there exists a constant C depending only on the constants in Assumptions.7 and. and which, in particular, is independent of, p K, h K and u, u S so that η K u S ) Cp 3/ ) K u us H;ωK + osc K. 4.7) Proof. We apply the results [16, Lemmata 3.4, 3.5]. There, the proofs are given for two space dimensions, i.e., d =. They carry over to the case d = 1 simply by using [16, Lemma.4] instead of [16, Theorem.5]. For the case d = 3, a careful inspection of the proofs of [16, Theorem.5] which is given in [13, Theorem D]) and [16, Lemma.6] shows that these lemmata also hold for d = 3. Hence, the proofs of [16, Lemmata 3.4, 3.5] can be used verbatim for the cases d = 1 and d = 3. We choose α = 0 in [16, Lemmata 3.4, 3.5]. Following these lines of arguments, we get for any ε > 0, K T, E E K, and αk resu S ) L K) Cε) p K u u S ) L K) + p 1+ε K α K u u S ) L K) + K)) osc α K Resu S ) L E) Cε)pε K p K u u S ) L ω K ) + p 1+ε K α K u u S ) L ω K ) + )) osc K. Hence, since local diameters and polynomial degrees are comparable, we have α K resu S ) L K) + α E Resu S ) L E) αk resu S ) L K) + Cα K Resu S ) L E) Cε)p K1 + p ε K ) u u S ) L ω K ) + 4pε 1 K K,+αK + u u S ) L ω K ) + pε 1 K osck). 4.8) For the special choice ε = 1/ and with condition 4.6) we finally get η Ku S ) Cp 3 K u us H;ω K + osc K). Remar a) It is possible to choose any ε > 0 in 4.8) with Cε) 1/ε). The factor p 3/ K in the estimate 4.7) then can be replaced by p1+ε, while condition 4.6) has the weaer form K,+ h K /p K p 1/ ε K for ε 1/). However, in view of p K log) we thin that this is of minor importance. b) Theorem 4.1 could be completed by a data saturation condition for the terms osc K in 4.7), which would then allow to bound η K u S ) directly by the error. p 3/ K

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