A residual a posteriori error estimate for the time-domain boundary element method

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A residual a posteriori error estimate for the time-domain boundary element method

Heiko Gimperlein, Ceyhun Özdemir, David Stark, Ernst P Stephan

To cite this version:

Heiko Gimperlein, Ceyhun Özdemir, David Stark, Ernst P Stephan. A residual a posteriori error estimate for the time-domain boundary element method. [Research Report] Heriot–Watt University.

2018. �hal-01970474�

A residual a posteriori error estimate for the time–domain boundary element method

Heiko Gimperlein^∗† Ceyhun ¨Ozdemir^‡ David Stark^∗ Ernst P. Stephan^‡

Abstract

This article investigates residual a posteriori error estimates and adaptive mesh re- finements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and acoustic boundary conditions which hold for a large class of discretizations. Efficiency of the error estimate is shown for a natural discretization of low order. Numerical examples confirm the theoretical results. The resulting adaptive mesh refinement procedures in 3drecover the adaptive convergence rates known for elliptic problems.

Mathematics Subject Classification: 65N38 (primary); 65M15; 35L67 (secondary)

Key words: boundary element method; a posteriori error estimates; adaptive mesh refine- ments; screen problems; wave equation.

1 Introduction

The efficient numerical treatment of boundary integral equations using adaptive mesh re- finement procedures has been extensively investigated for the numerical solution of homo- geneous elliptic problems in unbounded domains [3, 5]. See [17] for a recent exposition.

In this article we investigate the extension of the a posteriori error analysis and adap- tive mesh refinement procedures to initial-boundary value problems for the wave equation, formulated as boundary integral equations in the time-domain [8, 27]. We prove a reliable a posteriori error estimate of residual type for a large class of conforming discretizations.

It is efficient for a time-domain boundary element method on a globally quasi-uniform mesh. The error estimate defines an adaptive mesh refinement procedure, which recovers the convergence rates known for time-independent screen problems.

There has been recent interest in the solution of such problems on adapted meshes.

Similar to the elliptic case, singularities of the solution may appear at singular points of the boundary, as discussed in [19, 20, 21], and in trapping regions. Time-independent graded meshes have been shown to recover quasi-optimal convergence rates for edge and

∗Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot–Watt University, Edinburgh, EH14 4AS, United Kingdom, email: [email protected].

†Institute for Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany.

‡Institute of Applied Mathematics, Leibniz University Hannover, 30167 Hannover, Germany.

H. G. acknowledges partial support by ERC Advanced Grant HARG 268105 and the EPSRC Impact Accel- eration Account. C. ¨O. is supported by a scholarship of the Avicenna Foundation.

corner singularities [13]. First steps towards time-adaptivity are due to Sauter and Veit [25] in 2 dimensions, and also convolution quadrature methods with graded, non-adaptively chosen time steps have been studied, for example in [26]. Gl¨afke [16] showed first results towards space-time refinements in 2 dimensions, and in unpublished work Abboud uses ZZ error indicators for computations with space-adaptive mesh refinements for screen problems.

To describe the main results, we consider the wave equation

∂_t²u−∆u= 0 , u= 0 for t≤0,

in the complement of a polyhedral domain or screen Ω ⊂ R^d, with an emphasis on the challenging case d = 3. On the boundary Γ = ∂Ω both Dirichlet and acoustic boundary conditions,

u=f , respectively ∂_νu−α∂_tu=f ,

are considered. Here f is given,ν is the outer unit normal vector to Γ, and 0< α, α⁻¹ ∈ L^∞(Γ).

Following Bamberger and Ha Duong [1], we recast the boundary problem as a time dependent boundary integral equation. The Dirichlet problem is equivalent to a hyperbolic variant of Symm’s integral equation:

Vφ(t, x) = Z

R+×Γ

G(t−τ, x, y)φ(τ, y) dτ dsy =f(t, x) . (1) Here φ is sought in a space-time anisotropic Sobolev space H_σ¹(R⁺,He⁻¹²(Γ)), and G is a fundamental solution of the wave equation,

G(t−s, x, y) = H(t−s− |x−y|) 2πp

(t−s)²+|x−y|² in 2d, (2)

G(t−s, x, y) = δ(t−s− |x−y|)

4π|x−y| in 3d. (3)

HereHis the Heaviside function andδthe Dirac distribution. Our results apply, in particu- lar, to a Galerkin discretization of the weak form of (1) in a subspaceV ⊂H_σ¹(R⁺,He⁻¹²(Γ)),

Z

R+

Z

Γ

V∂tφ(t, x) ψ(t, x) dsxdσt= Z

R+

Z

Γ

∂tf(t, x) ψ(t, x)dsxdσt , (4) d_σt = e^−2σtdt with σ > 0. For computations we consider subspaces V = V_h,∆t^p,q of tensor products of piecewise polynomials in space and time, defined in Section 2. But also time discretizations based on smooth functions are of interest [25].

This article shows that norms of the residual give upper and lower bounds for the error.

The upper bound holds for arbitrary discretizations, not only the Galerkin method (4), and for general meshes:

Theorem A:Letφ∈H_σ¹(R⁺,He⁻¹²(Γ))be the solution to(4), and letφ_h,∆t∈H_σ¹(R⁺, H⁻¹²(Γ)) such thatR=∂_tf− V∂_tφ_h,∆t∈H_σ⁰(R⁺, H¹(Γ)). Then

kφ−φ_h,∆tk²_0,−1

2,Γ,∗ .X

i,∆

max{∆t, h∆} kRk²_{0,1,[ti,ti+1)×∆} . (5)

Let Γ be closed and polyhedral. For a globally quasi-uniform mesh on Γ, let V = W_h,∆t⁰ the tensor product of cubic splines in time with piecewise constant functions in space. If φ_h,∆t∈V is a Galerkin solution of (4) in V andφ∈H_σ²(R⁺, H⁻¹²(Γ)), then

max{∆t, h}kRk²_0,1−ǫ,Γ .kφ−φ_h,∆tk²_2,−1

2,Γ . (6)

The upper bound (5) is obtained in Corollary 4.5, the lower bound (6) in Theorems 5.1 and 6.1. Our numerical results illustrate the a posteriori error estimate of Theorem A for time-domain boundary elements based on (4), but the estimate also applies, for example, to convolution quadrature methods [27].

Note the loss of time derivatives between the upper and lower bound of the error, in the first Sobolev index. The loss is well-known for error estimates for hyperbolic problems [18].

Our arguments generalize to give reliable a posteriori estimates for the acoustic boundary problem, see Section 3.2.

The residual error estimate from Theorem A is used to define adaptive mesh refinements in space, based on the four steps Solve, Estimate, Mark, Refine. Numerical experiments confirm the efficiency and reliability of the estimate in examples. For screen problems they recover the convergence rates known for elliptic problems.

This work builds on the numerical analysis of adaptive boundary element methods for the Laplace equation, both for Symm’s integral equation and the hypersingular equation [3, 5, 6, 7]. Work on different types of error indicators in the time-independent case includes ZZ [4] and Faermann indicators [9, 10]. We also mention [31] for an earlier approach. A comparison of different indicators in the time-domain will be the subject of future research.

Structure of this article: Section 2 recalls the boundary integral operators associated to the wave equation as well as their mapping properties between suitable space-time anisotropic Sobolev spaces. The Sobolev spaces are discretized using tensor products of piecewise polynomials in space and time. Section 3 presents a corresponding space-time discretization for the formulation of the Dirichlet problem in terms of the single layer operator and derives a reliable a posteriori error estimate for globally quasi-uniform meshes, using a canonical approach which will readily adapt to other settings. A second subsection analyzes an acoustic boundary problem, a system of equations involving in addition the double layer, adjoint double layer and hypersingular operators. Section 4 then localizes the space-time Sobolev norm to derive the general upper estimate for the Dirichlet problem in Theorem A. The upper estimates are complemented by a lower bound for the error of a Galerkin approximation on globally quasi-uniform meshes in Section 5. The final step of this proof is a lower bound for the best approximation in Section 6. Section 7 discusses some algorithmic properties of the implementation, before numerical experiments are used to confirm the theoretical results in Section 8.

2 Preliminaries and discretization

In addition to the single layer operator V, for acoustic boundary problems we require its normal derivativeK^′, the double layer operatorK and hypersingular operatorW forx∈Γ, t >0:

Vϕ(t, x) = Z

R⁺×Γ

G(t−τ, x, y) ϕ(τ, y) dτ ds_y, (7) Kϕ(t, x) =

Z

R+×Γ

∂G

∂n_y(t−τ, x, y)ϕ(τ, y)dτ dsy , (8) K^′ϕ(t, x) =

Z

R⁺×Γ

∂G

∂n_x(t−τ, x, y) ϕ(τ, y) dτ ds_y, (9) Wϕ(t, x) =

Z

R⁺×Γ

∂²G

∂n_x∂n_y(t−τ, x, y) ϕ(τ, y) dτ ds_y . (10) Space–time anisotropic Sobolev spaces on the boundary Γ provide a convenient setting to study the mapping properties of layer potentials. See [18, 14] for a detailed exposition.

To define them, if∂Γ6=∅, first extend Γ to a closed, orientable Lipschitz manifold eΓ.

On Γ one defines the usual Sobolev spaces of supported distributions:

He^s(Γ) ={u∈H^s(eΓ) : suppu⊂Γ}, s∈R. Furthermore,H^s(Γ) is the quotient spaceH^s(eΓ)/He^s(eΓ\Γ).

To write down an explicit family of Sobolev norms, introduce a partition of unityαi subor- dinate to a covering ofΓ by open setse B_i. For diffeomorphismsϕ_i mapping eachB_iinto the unit cube⊂Rⁿ, a family of Sobolev norms is induced fromRⁿ, with parameterω∈C\ {0}:

||u||_s,ω,Γe= Xp

i=1

Z

Rⁿ

(|ω|²+|ξ|²)^s|F

(αiu)◦ϕ⁻_i ¹ (ξ)|²dξ

!¹₂ .

The norms for different ω ∈ C\ {0} are equivalent and F denotes the Fourier trans- form. They induce norms on H^s(Γ), ||u||s,ω,Γ = inf_v∈Hes(eΓ\Γ) ||u+v||_s,ω,Γe and on He^s(Γ),

||u||s,ω,Γ,∗ =||e+u||_s,ω,eΓ. We writeH_ω^s(Γ) forH^s(Γ), respectivelyHe_ω^s(Γ) forHe^s(Γ), when a norm with a specificωis fixed. e₊extends the distributionuby 0 from Γ toΓ. As the norme

||u||s,ω,Γ,∗ corresponds to extension by zero, while||u||s,ω,Γ allows extension by an arbitrary v,||u||_s,ω,Γ,∗ is stronger than||u||_s,ω,Γ. Like in the time-independent case the norms are not equivalent whenevers∈ ¹₂+Z[17].

We now define a class of space-time anisotropic Sobolev spaces:

Definition 2.1. For σ >0 and r, s∈Rdefine

H_σ^r(R⁺, H^s(Γ)) ={u∈ D₊^′ (H^s(Γ)) :e^−σtu∈ S₊^′ (H^s(Γ)) and ||u||r,s,σ,Γ<∞}, H_σ^r(R⁺,He^s(Γ)) ={u∈ D₊^′ (He^s(Γ)) :e^−σtu∈ S₊^′ (He^s(Γ)) and ||u||r,s,σ,Γ,∗ <∞}. D₊^′ (E) respectively S₊^′ (E) denote the spaces of distributions, respectively tempered distri- butions, on R with support in [0,∞), taking values in E = H^s(Γ),He^s(Γ). The relevant norms are given by

kukr,s,σ:=kukr,s,σ,Γ =

Z +∞+iσ

−∞+iσ

|ω|^2r kˆu(ω)k²_s,ω,Γ dω ¹₂

,

kukr,s,σ,∗ :=kuk_{r,s,σ,Γ,∗} =

Z +∞+iσ

−∞+iσ

|ω|^2r kˆu(ω)k²_s,ω,Γ,∗ dω ¹₂

.

They are Hilbert spaces, and we note that the basic case r = s = 0 is the weighted L²-space with scalar product hu, vi_σ :=R_∞

0 e^−2σtR

Γuvds_x dt. Because Γ is Lipschitz, like in the case of standard Sobolev spaces these spaces are independent of the choice ofα_i and ϕ_i when |s| ≤1.

Using variational arguments, precise mapping properties are well-known for the layer po- tentials between Sobolev spaces related to the energy:

Theorem 2.2 ([14]). The following operators are continuous for r∈R: V :H_σ^r+1(R⁺,He⁻¹²(Γ))→H_σ^r(R⁺, H¹²(Γ)) , K^′ :H_σ^r+1(R⁺,He⁻¹²(Γ))→H_σ^r(R⁺, H⁻¹²(Γ)) ,

K:H_σ^r+1(R⁺,He¹²(Γ))→H_σ^r(R⁺, H¹²(Γ)) , W :H_σ^r+1(R⁺,He¹²(Γ))→H_σ^r(R⁺, H⁻¹²(Γ)) . More generally, outside the energy spaces one has:

Theorem 2.3 ([16]). The following operators are continuous for r∈R, s∈(−¹₂,¹₂):

V :H_σ^r+1(R⁺,He^−12+s(Γ))→H_σ^r(R⁺, H^12+s(Γ)) , K^′:H_σ^r+2(R⁺,He^−12+s(Γ))→H_σ^r(R⁺, H^−12+s(Γ)) ,

K:H_σ^r+2(R⁺,He^12+s(Γ))→H_σ^r(R⁺, H^12+s(Γ)) , W :H_σ^r+3(R⁺,He^12+s(Γ))→H_σ^r(R⁺, H^−12+s(Γ)) .

For simplicity, we will use here a hypersurface composed of N_s triangular facets such that Γ =∪^N_i=1^s T_i, each elementT_i is closed withint(T_i)6=∅, and for distinctT_i, T_j ⊂Γ the intersectionint(Ti)∩int(Tj) =∅.

For the time discretization we consider a uniform decomposition of the time intervalR⁺ into subintervalsIn= [tn−1, tn) with time step|In|= ∆t, such thattn=n∆t,n= 0,1, . . .. We choose a basis {ϕ^p_j} of the space V_h^p of piecewise polynomial functions of degree p in space (continuous if p ≥ 1) and a basis {β^j,q} of the space V_∆t^q of piecewise polynomial functions of degree ofq in time (continuous and vanishing att= 0 if q≥1). In addition to V_∆t^q we also require the spaceW_∆t⊂V_∆t³ of cubic splines.

We denote the finite temporal mesh{[0, t1),[t₁, t₂),· · · ,[t_Nt−1, t_Nt)}byTT, the spatial mesh {T1,· · ·, T_Ns} for Γ byT_S.

We consider the tensor product of the approximation spaces in space and time,V_h^p andV_∆t^q , associated to the space–time meshTS,T =TS× TT, and we write

V_h,∆t^p,q =V_h^p⊗V_∆t^q .

We consider the orthogonal projections Π_∆t from L²(R₊) to V_∆t^q , resp. Π_h from L²(Γ) to V_h^p. See [14] for a discussion of their properties and those of their composition Πe_h,∆t. Furthermore, we defineW_h,∆t^p =V_h^p⊗W_∆t.

One obtains as in Proposition 3.54 of [16]:

Lemma 2.4. Let f ∈H_σ^r(R⁺, H^m(Γ)∩He^s(Γ)), 0< m≤q+ 1, 0< r≤p+ 1, s≤r, |l| ≤ ¹₂ such thatls≥0. Then if l, s≤0

kf −Π_h◦Π_∆tfks,l,Γ ≤C(h^α+ (∆t)^β)||f||r,m,Γ ,

where α = min{m−l, m− ^m(l+s)_m+r }, β = min{m+r−(l+s), m+r− ^m+r_m l}. If l, s > 0, β=m+r−(l+s).

For the lower bounds, we further require an auxiliary projectionP_h,∆tfromH²(R⁺, H⁻¹(Γ)) into the space of ansatz functions⊆H²(R⁺, L²(Γ)) such thatP_h,∆tisH²(R⁺, L²(Γ))-stable.

For the construction of P_h,∆t = P_∆tQ^∗_h we use a projection P_∆t on the space of cubic splines W_∆t⊂V_∆t³ in time and a Galerkin projectionQ^∗_h in space.

More precisely, we consider the projection Qh : L²(Γ) → V_h^p defined as follows: For φ∈L²(Γ), we letQ_hφ∈V_h^p be the unique solution to the variational problem

hQhφ, ψ_hi=hφ, ψhi

for all ψ_h ∈ V_h^p. Q_h is bounded on L²(Γ) and self-adjoint, Q_h = Q^∗_h. Therefore, if Q_h restricts to a bounded operator on H_ω^s(Γ), Qh = Q^∗_h extends with the same norm to a bounded operator onHe_ω^−s(Γ).

If the triangulation is globally quasi-uniform, we show that Q_h is a bounded operator onH_ω¹(Γ) with operator norm uniformly bounded in ω:

Lemma 2.5. Assume that Γ does not have reentrant corners. Consider a globally quasi- uniform mesh and s∈[0,1]. For all φ∈H_ω^s(Γ), we have:

kQ_hφk_s,ω,Γ≤Ckφk_s,ω,Γ. For all φ∈He_ω^−s(Γ), we have:

kQ_hφk_{−s,ω,Γ,∗}≤Ckφk_{−s,ω,Γ,∗}.

Proof. For the first assertion we note that it is clear for s= 0. We show it for s= 1, and the case for generals∈[0,1] follows from interpolation.

LetQ_h,1 :H_ω¹(Γ)→V_h^p ⊂H_ω¹(Γ) defined by

hQh,1φ, ψ_hi1,ω,Γ=hφ, ψhi1,ω,Γ

for allψ_h ∈V_h^p. By definition, Q_h,1 is bounded on H_ω¹(Γ). We use the Aubin-Nitsche trick to show that

kφ− Qh,1φk_L²_(Γ).hkφk_H1 ω(Γ) .

More precisely, ifg∈L²(Γ) denote by φ_g ∈H_ω¹(Γ) the solution to the elliptic equation hψ, φgi1,ω,Γ=hg, ψi_L²_(Γ) ,

for allψ∈H_ω¹(Γ) . By elliptic regularityφ_g ∈H_ω²(Γ), withkφ_gk_2,ω,Γ .kgk_L²_(Γ), as Γ does not have reentrant corners. We observe

kφ− Qh,1φk_L²_(Γ)= sup

06=g∈L²(Γ)

hg, φ− Qh,1φi_L²_(Γ) kgk_L²_(Γ)

= sup

06=g∈L²(Γ)

hφ− Qh,1φ, φ_gi1,ω,Γ

kgk_L²_(Γ)

= sup

06=g∈L²(Γ)

06=ψinfh∈V_h^p

hφ− Qh,1φ, φ_g−ψ_hi1,ω,Γ

kgk_L2(Γ)

≤ kφ− Q_h,1φk_1,ω,Γ sup

06=g∈L²(Γ)

inf

06=ψh∈V_h^p

kφg−ψ_hk1,ω,Γ

kgk_L²_(Γ) .

Becauseφ_g ∈H_ω²(Γ), inf

06=ψh∈V_h^pkφg−ψ_hk1,ω,Γ.hk∇Γφ_gk1,ω,Γ.hkgk_L²_(Γ) andkφ− Qh,1φk1,ω,Γ .kφk1,ω,Γ, we conclude

kφ− Qh,1φk_L²_(Γ) .hkφk1,ω,Γ .

Combining this with the triangle inequality, the inverse inequality and the already shown assertion whens= 0, we obtain the first assertion:

kQhφk1,ω,Γ≤ kQh,1φk1,ω,Γ+kQhφ− Qh,1φk1,ω,Γ

.kφk_1,ω,Γ+h⁻¹kQ_hφ− Q_h,1φk_L2(Γ)

=kφk_1,ω,Γ+h⁻¹kQ_h(φ− Q_h,1φ)k_L²_(Γ) .kφk_1,ω,Γ+h⁻¹kφ− Q_h,1φk_L²_(Γ) .kφk1,ω,Γ .

The second assertion follows from the first one, with the same constantC. Indeed, kQhφk₋s,ω,Γ,∗ = sup

06=ψ∈H_ω^s(Γ)

hQhφ, ψi_L²_(Γ) kψks,ω,Γ

= sup

06=ψ∈H_ω^s(Γ)

hφ,Q_hψi_L²_(Γ)

kψk_s,ω,Γ ≤Ckφk₋s,ω,Γ,∗ .

Using the Laplace transform to transfer the result into the time-domain, we obtain kQhφkr,−s,Γ,∗ ≤Ckφkr,−s,Γ,∗

fors∈[0,1] and all r. Further note that the interpolation operator P_∆t in time for cubic splines is continuous on H² functions. The continuity extends to H²(R⁺,He^−s(Γ)). We conclude:

Lemma 2.6. Consider a quasi-uniform mesh, s ∈ [0,1] and P_h,∆t = P∆tQ_h. For all φ∈H²(R⁺,He^−s(Γ)), we have

kPh,∆tφk2,−s,Γ,∗≤Ckφk2,−s,Γ,∗.

The approximation properties ofP_h,∆tanalogous to Lemma 2.4 follow as in [16], Propo- sition 3.54.

3 A posteriori error estimates – reliability

3.1 Dirichlet problem

We recall the basic properties of the bilinear form BD(φ, ψ) =

Z

R+

Z

Γ

V∂tφ(t, x)ψ(t, x)dsxdσt of the Dirichlet problem.

As shown in [18], the bilinear form is continuous, and also weakly coercive:

Proposition 3.1. For every φ, ψ∈H_σ¹(R⁺, H⁻¹²(Γ)) there holds:

|BD(φ, ψ)|.kφk_1,−¹

2,Γ,∗kψk_1,−¹

2,Γ,∗

and

kφk²_0,−1

2,Γ,∗ .B_D(φ, φ).

We consider a conforming Galerkin discretization of the Dirichlet problem (4) in a subspaceV ⊂H_σ¹(R⁺,He⁻¹²(Γ)), which reads as follows: Findφ_h,∆t∈V such that

B_D(φ_h,∆t, ψ_h,∆t) =h∂tf, ψ_h,∆ti , (11) for allψ_h,∆t∈V.

The well-posedness of the continuous and discretized problems are a basic consequence of Proposition 3.1:

Corollary 3.2. Let f ∈H_σ²(R⁺, H¹²(Γ)). Then the Dirichlet problem (4)and its discretiza- tion (11) admit unique solutionsφ∈H_σ¹(R⁺,He⁻¹²(Γ)), φ_h,∆t∈V, and the estimates

kφk_1,−¹

2,Γ,∗,kφh,∆tk_1,−¹

2,Γ,∗ .kfk_2,¹

2,Γ

hold.

We note the Galerkin orthogonality:

BD(φ−φ_h,∆t, ψ_h,∆t) = 0 ∀ψ_h,∆t∈V .

Using ideas going back to Carstensen [3] and Carstensen and Stephan [5] for the bound- ary element method for elliptic problems, we obtain an a posteriori error estimate for the Galerkin solution to the Dirichlet problem on globally quasi-uniform meshes.

Theorem 3.3. Given V = V_h,∆t^p,q associated to a globally quasi-uniform discretization of Γ, let φ ∈ H_σ¹(R⁺,He⁻¹²(Γ)), φ_h,∆t ∈ V the solutions to (4), resp. (11). Assume that R=∂_tf − V∂tφ_h,∆t∈H_σ⁰(R⁺, H¹(Γ)). Then

kφ−φ_h,∆tk²_0,−1

2,Γ,∗ .kRk0,1,Γ ∆tk∂tRk0,0,Γ+kh· ∇Rk0,0,Γ .max{∆t, h}(k∂_tRk²_0,0,Γ+k∇Rk²_0,0,Γ) . Proof of Theorem 3.3. We first note that for allψ_h,∆t∈V_h,∆t^p,q

kφ−φ_h,∆tk²_0,−1

2,Γ,∗ .BD(φ−φ_h,∆t, φ−φ_h,∆t)

= Z

R⁺

Z

Γ

∂_tf(φ−φ_h,∆t) ds_x d_σt−B_D(φ_h,∆t, φ−φ_h,∆t)

= Z

R⁺

Z

Γ

∂_tf(φ−ψ_h,∆t)ds_x d_σt−B_D(φ_h,∆t, φ−ψ_h,∆t)

= Z

R⁺

Z

Γ

(∂_tf− V∂_tφ_h,∆t)(φ−ψ_h,∆t)ds_x d_σt .

The last term may be estimated by:

Z

R⁺

Z

Γ

(∂_tf − Vφ˙_h,∆t)(φ−ψ_h,∆t) ds_x d_σt

≤ kRk_0,¹

2,Γkφ−ψ_h,∆tk_0,−¹

2,Γ,∗ . We useψ_h,∆t=φ_h,∆t together with the interpolation inequality

kRk²_0,1

2,Γ≤ kRk_0,0,ΓkRk_0,1,Γ . As the residual is perpendicular toV_h,∆t^p,q ,

kRk²_0,0,Γ=hR,Ri=hR,R −ψe_h,∆ti

≤ kRk0,0,ΓkR −ψe_h,∆tk0,0,Γ

for allψe_h,∆t∈V_h,∆t^p,q , we obtain

kRk0,0,Γ ≤inf{kR −ψe_h,∆tk0,0,Γ :ψe_h,∆t∈V_h,∆t^p,q }.

Choosingψe_h,∆t=Πe_h,∆tR, based on the interpolation operator defined earlier, we obtain kRk0,0,Γ .∆tk∂tRk0,0,Γ+kh· ∇Rk0,0,Γ .

The theorem follows.

3.2 Acoustic boundary problems

Recall the wave equation with inhomogeneous acoustic boundary conditions

∂_t²u−∆u= 0 onR₊×Ω , ∂νu−α∂tu=f onR₊×Γ , u= 0 for t≤0. For scattering problemsf =−∂νu_inc+α∂_tu_inc is determined from an incoming waveu_inc.

For a finite or infinite time interval [0, T] we introduce the bilinear form a_T((ϕ, p),(ψ, q)) =

Z T

0

Z

Γ

αϕ˙ψ˙+ 1

αpq+K^′pψ˙− Wϕψ˙+Vpq˙ −Kϕq˙

ds_xdt . (12) With

l(ψ, q) = Z T

0

Z

Γ

Fψ ds˙ _xdt+ Z _∞

0

Z

Γ

Gq

α ds_xdt , (13)

where F = −2∂νu_inc, G= 2α∂_tu_inc, we consider the variational formulation for the wave equation in R³ with acoustic boundary conditions on Γ:

Find (ϕ, p)∈H¹([0, T],He¹²(Γ))×H¹([0, T], L²(Γ)) such that

a_T((ϕ, p),(ψ, q)) =l(ψ, q) (14)

for all (ψ, q)∈H¹([0, T], H¹²(Γ))×H¹([0, T], L²(Γ)).

Note thatσ may be set to 0 in the definition of the Sobolev spaces whenT < ∞. The acoustic problem is equivalent to the wave equation with acoustic boundary conditions [18].

Its discretization reads:

Find (ϕ_h,∆t, p_h,∆t)∈V_h,∆t^p1,q1 ×V_h,∆t^p2,q2 such that

aT((ϕ_h,∆t, p_h,∆t),(ψ_h,∆t, q_h,∆t)) =l(ψ_h,∆t, q_h,∆t) (15) for all (ψ_h,∆t, q_h,∆t)∈V_h,∆t^p1,q1×V_h,∆t^p2,q2.

The following well-posedness holds:

Proposition 3.4. Let F ∈ H²([0, T], H⁻¹²(Γ)), G ∈ H¹([0, T], H⁰(Γ)). Then the weak form(14) of the acoustic problem and its discretization (15)admit unique solutions(ϕ, p)∈ H¹([0, T],He¹²(Γ))×H¹([0, T], L²(Γ)), resp. (ϕ_h,∆t, p_h,∆t) ∈ V_h,∆t^p1,q1 ×V_h,∆t^p2,q2, which depend continuously on the data.

We specifically note that the bilinear form a_T satisfies a (weaker) coercivity estimate:

kpk²_0,0,Γ+kϕk˙ ²_0,0,Γ.aT((ϕ, p),(ϕ, p)). This follows from Equation (64) of [18],

a_T((ϕ, p),(ϕ, p)) = 2E(T) + Z T

0

Z

Γ

α(∂_tϕ)(∂_tϕ) + 1 αpq

ds_xdt ,

whereE(T) = ¹₂R

R^d\Ω

(∂tu)²+ (∇u)² dxis the total energy at timeT. We state a simple a posteriori estimate.

Theorem 3.5. Let (ϕ, p) ∈ H¹([0, T],He¹²(Γ))×H¹([0, T], L²(Γ)) be the solution to (14), (ϕ_h,∆t, p_h,∆t)∈V_h,∆t^p1,q1 ×V_h,∆t^p2,q2 the solution to the discretization (15). Assume that

R₁ =F−αϕ˙_h,∆t+ 2K^′p_h,∆t−2Wϕ_h,∆t∈L²([0, T], L²(Γ)) , R2 =G+α⁻¹p_h,∆t+ 2Vp˙_h,∆t−2Kϕ˙_h,∆t∈L²([0, T], L²(Γ)) . Then

kp−p_h,∆tk0,0,Γ+kϕ˙−ϕ˙_h,∆tk0,0,Γ.kR1k0,0,Γ+kR2k0,0,Γ . Proof. For every (ψ_h,∆t, q_h,∆t)∈V_h,∆t^p1,q1 ×V_h,∆t^p2,q2 we have

kp−p_h,∆tk²_0,0,Γ+kϕ˙−ϕ˙_h,∆tk²_0,0,Γ

.aT((ϕ−ϕ_h,∆t, p−p_h,∆t),(ϕ−ϕ_h,∆t, p−p_h,∆t))

=hR1,ϕ˙−ψ˙_h,∆ti+hR2, p−q_h,∆ti

≤ kR1k0,0,Γkϕ˙−ψ˙_h,∆tk0,0,Γ

+kR2k0,0,Γkp−q_h,∆tk0,0,Γ

≤(kR₁k_0,0,Γ+kR₂k_0,0,Γ)×

(kp−q_h,∆tk0,0,Γ+kϕ˙−ψ˙_h,∆tk0,0,Γ) . The assertion is obtained by choosing (ψ_h,∆t, q_h,∆t) = (ϕ_h,∆t, p_h,∆t).

Naturally, for a quasi-uniform discretization of Γ and under stronger assumptions on R₁, R₂we may obtain powers ofhand ∆ton the right hand side by the following argument:

As in the proof of Theorem 3.3,hR1,ψe˙_h,∆ti=hR2,eq_h,∆ti= 0 for all (ψe_h,∆t,eq_h,∆t)∈V_h,∆t^p1,q1× V_h,∆t^p2,q2. Hence

kR2k²_0,0,Γ=hR2, R₂i=hR2, R₂−qe_h,∆ti

≤ kR2k0,0,ΓkR2−eq_h,∆tk0,0,Γ. Choosingqe_h,∆t= Π_h,∆tR₂ yields

kR2k0,0,Γ.∆tk∂tR₂k0,0,Γ+kh· ∇ΓR₂k0,0,Γ+ ∆tkh· ∇Γ∂_tR₂k0,0,Γ

providedR₂ ∈H¹([0, T], H¹(Γ)).

AssumingR1 ∈H¹([0, T], H¹(Γ)), we similarly have

kR1k²_0,0,Γ=hR1, R₁i=hR1, R₁−ψ˙e_h,∆ti

≤ kR1k¹

2−s,0,Γk Z t

0

R₁−ψe_h,∆tk¹

2+s,0,Γ . Choosingψe_h,∆t= Π_h,∆tRt

0R₁ and s= ¹₂ results in

kR1k0,0,Γ .∆tk∂tR₁k0,0,Γ+kh· ∇ΓR₁k0,0,Γ

+ ∆tkh· ∇Γ∂_tR₁k0,0,Γ . Altogether,

kp−p_h,∆tk0,0,Γ+kϕ˙−ϕ˙_h,∆tk0,0,Γ+kϕ−ϕ_h,∆tk_0,¹

2,Γ

.kR1k0,0,Γ+kR2k0,0,Γ

. X2

i=1

∆tk∂_tR_ik_0,0,Γ+kh· ∇R_ik_0,0,Γ + ∆tkh· ∇∂tR_ik0,0,Γ .

4 Error estimates for general discretizations

This section generalizes the results for the single layer potential without any assumptions on the underlying meshes.

We recall Lemma 3 in [13]:

Lemma 4.1. Let f₁, . . . , f_n ∈ H_σ^r(R⁺,He^s(Γ)), 0 ≤ s ≤ 1, r ≥ 0, such that f_jf_k = 0 for j6=k. Let ω_j be the interior of the support of f_j with ω_j = supp f_j. Then

Xn

j=1

fj

²

r,s,Γ,∗ ≤ Xn

j=1

kfjk²_r,s,ω

j,∗ .

The lemma will be applied to a finite partition of unity Φ given by non-negative Lipschitz functions{φj}^M_j=1 on Γ such that PM

j=1φ_j = 1.

Definition 4.2. The overlap of the partition of unity Φ is defined asK(Φ) = max_jcard{k: φ_kφj 6= 0}.

For a partition of unity associated to a triangulation, M tends to infinity as the mesh size decreases, while the overlap may be much smaller. We note a crucial observation from [6], Lemma 3.1:

Lemma 4.3. Let Φ be a finite partition of unity of ∂Ω with overlap K(Φ). Then there exists a partition of {1, . . . , M} into K ≤K(Φ)non-empty subsets M₁, . . . , M_K, such that SK

j=1Mj ={1, . . . , M},Mj∩Mk =∅ ifj6=kand for all l∈ {1, . . . , K}and j, k∈Ml with j6=k, φ_jφ_k= 0 on[0, T]×∂Ω.

Similar to the partition of unity in space, associated to the decomposition of the time axis,R⁺=S_∞

l=0[l∆t,(l+1)∆t), we consider a partition of unity{ψl}^∞_l=0onR⁺,P_∞

l=0ψ_l = 1.

We may choose it so thatψ_lis supported in the interval ((l−¹₂)∆t,(l+³₂)∆t), and therefore ψ_lψ_l^′ = 0 whenever l∈Mf₀= 2N,l^′ ∈Mf₁ = 2N+ 1.

We obtain a partition of unity{ψe_l,j=ψ_l⊗φ_j}l,j on R⁺×∂Ω.

Theorem 4.4. Let Γ^′ ⊂Γbe connected. and letΦbe a finite partition of unity with overlap K(Φ). Then for any f ∈H_σ^r(R⁺,He^s(Γ^′)) and any 0≤s≤1, we have

kfk²_r,s,Γ^′_,∗ ≤2K(Φ) X∞ l=0

XM

j=1

kfψel,jk²⁽¹_r,0,Γ⁻′^s)kfψel,jk^2s_r,1,Γ^′ .

Proof. We show

kfk²_r,s,Γ^′_,∗≤2K(Φ) X∞

l=0

XM

j=1

kfψe_l,jk²_r,s,Γ^′_,∗ . (16) The assertion then follows from the interpolation estimate

kfψe_l,jkr,s,Γ^′,∗ .kfψe_l,jk¹_r,0,Γ^−s′kfψe_l,jk^s_r,1,Γ^′ .

To show (16), we consider a partitionM₁, . . . , M_Kas in Lemma 4.3. Thenf =P_∞

l=0

PM

j=1ψe_l,jf, so that

kfk²_r,s,Γ^′_,∗ =k X1

m=0

X

l∈Mfm

XK

k=1

X

j∈Mk

ψe_l,jfk²_r,s,Γ^′_,∗ ≤2K X1

m=0

XK

k=1

k X

l∈Mfm

X

j∈Mk

ψe_l,jfk²_r,s,Γ^′_,∗ .

With Lemma 4.1 andI_l= ((l−1)∆t,(l+ 2)∆t), k X

l∈Mfm

X

j∈Mk

ψe_l,jfk²_r,s,Γ′,∗ ≤ X

j∈Mk

k X

l∈Mfm

ψe_l,jfk²_r,s,ωj,∗ ≤ X

l∈Mfm

X

j∈Mk

kψe_l,jfk²_{r,s,Il×ωj,∗} ,

so that

kfk²_r,s,Γ^′_,∗ ≤2K X1

m=0

X

l∈Mfm

XK

k=1

X

j∈Mk

kψe_l,jfk²_{r,s,Il×ωj,∗} = 2K X∞

l=0

XM

j=1

kψe_l,jfk²_{r,s,Il×ωj,∗} ,

which shows (16).

Together with Friedrichs inequality

kfψe_l,jk_r,1,Γ^′ .σ k∇(fψe_l,j)k_r,0,Γ^′+k∂t(fψe_l,j)k_r,0,Γ^′

one obtains

kfψe_l,jk¹_r,0,Γ^−s′kfψe_l,jk^s_r,1,Γ^′ .σ d²⁽¹_j ^−s)(1 +d²_j)^s

k∇(fψe_l,j)k²_r,0,Γ^′ +k∂t(fψe_l,j)k²_r,0,Γ^′ .

Hered_j = max{δj,∆t}, where δ_j is the width of the support of φ_j, defined as the smallest number such that the following is true: There exists a direction n ∈ R³, |n| = 1, such that for allx ∈ R³ and for each planeH perpendicular to n with x∈ H, the intersection suppφ_j∩H is a Lipschitz curve of length≤δ_j.

We now useV∂t(φ−φ_h,∆t) =R, the coercivity ofV∂_t in Proposition 3.1, and Theorem 4.4 withs= ¹₂. The following a posteriori error estimate follows:

Corollary 4.5. Letφ∈H_σ¹(R⁺, H⁻¹²(Γ))be the solution to(4), and letφ_h,∆t∈H_σ¹(R⁺, H⁻¹²(Γ)) such thatR=∂tf− V∂tφ_h,∆t∈H_σ⁰(R⁺, H¹(Γ)). Then

kφ−φh,∆tk²_0,−1 2,Γ,∗.σ

X

j

δ_j

k∇Rk²_0,0,Il×∆j+k∂tRk²_0,0,Il×∆j

≃X

l,∆j

max{∆t, h∆} kRk²_0,1,Il×∆j .

5 Lower bounds

As for time-independent problems, for the discussion of lower bounds we restrict ourselves to globally quasi-uniform meshes.

Because of the different norms in the upper and lower bounds for B_D in Proposition 3.1, the a posteriori estimate only satisfies a weak variant of efficiency:

Providedφ∈H_σ²(R⁺, H⁰(Γ)) we have forε∈(0,1):

max{∆t, h}^−1−ε2 kφ−φ_h,∆tk_0,−¹

2,Γ,∗.

kRk0,1−ε,Γ=kV( ˙φ−φ˙_h,∆t)k0,1−ε,Γ.kφ−φ_h,∆tk2,−ε,Γ≤ kφ−φ_h,∆tk2,0,Γ . A proof of the sharp estimate, ε= 0, would require sharp mapping properties of the layer potentials outside the energy spaces and will be pursued elsewhere.

As in the elliptic case, we aim to use the mapping properties ofV together with approx- imation properties of the finite element spaces to recover the same spatial Sobolev index

−¹₂ in the upper and lower estimates.

Theorem 5.1. Assume that R ∈ H_σ⁰(R⁺, H¹(Γ)) and that the ansatz functions V = W_h,∆t^p ⊂H_σ²(R⁺, H⁰(Γ))∩V_h,∆t^p,q satisfy

ψh∆tinf∈V kφ−ψ_h∆tk2,0,Γ≃max{h,∆t}^β (17) for some β >0. Then for allε∈(0,1)

kRk0,1−ε,Γ.max{h⁻¹²,(∆t)⁻¹²}kφ−φ_h∆tk_2,−¹

2,Γ.

Remark 5.2. In Theorem 6.1 below, the hypothesis (17) is verified using the singular expansion of the solutionφ at the edges and corners.

Proof of Theorem 5.1. The theorem partly follows the functional analytic approach of [3]

for the time-independent case. We consider the following quantities: the error of the best approximation Π_h,∆tφof φinH_σ²(R⁺, L²(Γ)),

E(φ, h,∆t) =kφ−Π_h,∆tφk2,0,Γ ,

whereE(φ, h,∆t)≃max{h,∆t}^β by (17); the relative error of best approximation, F(φ, h,∆t) = kφ−Π_h,∆tφk2,−1,Γ

kφ−Π_h,∆tφk2,0,Γ

; and the inverse inequality [16]

G(h,∆t) = sup

ψh∆t6=0

kψh∆tk2,0,Γ

kψ_h∆tk_2,−1,Γ .max{h⁻¹,∆t⁻¹}.

Their analysis relies on the projection operatorP_h,∆t from Lemma 2.6.

Note that F(φ, h,∆t) . max{h,∆t}. Indeed because φ ∈ H_σ²(R⁺, H⁰(Γ)), by duality and the approximation properties of Π_h,∆t:

kφ−Π_h,∆tφk2,−1,Γ= sup

η

hφ−Π_h,∆tφ, η−Π_h,∆tηi kηk₋2,1,Γ

.max{h,∆t}kφ−Π_h,∆tφk2,0,Γ. This uses the estimatekη−Π_h,∆tηk₋2,0,Γ.max{h,∆t}kηk₋2,1,Γ.

From the beginning of this section, we recall

kRk0,1−ε,Γ .kφ−φ_h,∆tk2,0,Γ ≤ kφ−Π_H,∆Tφk2,0,Γ+kΠH,∆Tφ−φ_h,∆tk2,0,Γ . Note that the first term is kφ−Π_H,∆Tφk2,0,Γ = E(φ, H,∆T). For the second term we observe that

kΠ_H,∆Tφ−φ_h,∆tk_2,0,Γ=kP_H,∆T(Π_H,∆Tφ−φ_h,∆t)k_2,0,Γ .G(H,∆T)kΠH,∆Tφ−φh,∆tk2,−1,Γ , so thatkΠH,∆Tφ−φ_h,∆tk2,0,Γ .G(H,∆T)^1/2kΠH,∆Tφ−φ_h,∆tk_2,−¹

2,Γ. We conclude kRk_0,1−ε,Γ .E(φ, H,∆T) +G(H,∆T)^1/2kΠ_H,∆Tφ−φ_h,∆tk_2,−¹

2,Γ . Further,

kΠH,∆Tφ−φ_h,∆tk_2,−¹

2,Γ≤ kΠH,∆Tφ−φk_2,−¹

2,Γ+kφ−φ_h,∆tk_2,−¹

2,Γ . From the definition ofF and interpolation, kΠ_H,∆Tφ−φk_2,−¹

2,Γ is bounded by .F(φ, H,∆T)^1/2kΠH,∆Tφ−φk2,0,Γ=F(φ, H,∆T)^1/2E(φ, H,∆T) . To sum up,

kφ−φ_h,∆tk2,0,Γ .E(φ, H,∆T) +G(H,∆T)^1/2F(φ, H,∆T)^1/2E(φ, h,∆t) +G(H,∆T)^1/2kφ−φ_h,∆tk_2,−¹

2,Γ

≤ E(φ, H,∆T)

E(φ, h,∆t) kφ−φ_h,∆tk_2,0,Γ(1 +G(H,∆T)^1/2F(φ, H,∆T)^1/2) +G(H,∆T)^1/2kφ−φ_h,∆tk_2,−1

2,Γ ,