A PLANE WAVE DISCONTINUOUS GALERKIN METHOD WITH A DIRICHLET-TO- NEUMANN BOUNDARY CONDITION FOR A SCATTERING PROBLEM IN ACOUSTICS

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1 A PLAE WAVE DISCOTIUOUS GALERKI METHOD WITH A DIRICHLET-TO- EUMA BOUDARY CODITIO FOR A SCATTERIG PROBLEM I ACOUSTICS By Selvean Kapita and Peter Monk IMA Preprint Series #474 December 06) ISTITUTE FOR MATHEMATICS AD ITS APPLICATIOS UIVERSITY OF MIESOTA 400 Lind Hall 07 Curc Street S.E. Minneapolis, Minnesota Pone: Fax: URL: ttp://

2 A Plane Wave Discontinuous Galerkin Metod wit a Diriclet-to-eumann Boundary Condition for te Scattering Problem in Acoustics Selvean Kapita, Peter Monk. Abstract We consider te numerical solution of an acoustic scattering problem by te Plane Wave Discontinuous Galerkin Metod PWDG) in te exterior of a bounded domain in R. In order to apply te PWDG metod, we introduce an artificial boundary to truncate te domain, and we impose a non-local Diriclet-to-eumann Dt) boundary condition on te artificial curve. To define te metod, we introduce new consistent numerical fluxes tat incorporate te truncated series of te Dt map. Error estimates wit respect to te truncation order of te Dt map, and wit respect to mes widt are derived. umerical results suggest tat te accuracy of te PWDG metod for te acoustic scattering problem can be significantly improved by using Dt boundary conditions. Contents Introduction Te Scattering Problem 3. on-local Boundary Value Problem Stability of te adjoint problem Some properties of Hankel functions Truncated Boundary Value Problem Stability of te truncated adjoint problem Te Plane Wave Discontinuous Galerkin Metod 9 4 Error Estimates 3 4. A mes-dependent error estimate Error estimates in te L norm Estimation of u u L Ω) Estimation of u u L Ω) umerical Implementation 6 umerical Results 4 7 Conclusion 9 Institute for Matematics and Its Applications, University of Minnesota, 07 Curc St SE, Minneapolis, Minnesota, Department of Matematical Sciences, University of Delaware, 5 Orcard Rd, ewark, Delaware, 976

3 8 Acknowledgements 9. Introduction Acoustic, elastic and electromagnetic scattering problems arise in many areas of pysical and engineering interest, in areas as diverse as radar, sonar, building acoustics, medical and seismic imaging. Matematically, te problem of acoustic scattering is often modeled by te Helmoltz equation in te unbounded region exterior to a bounded obstacle Ω D R. Te main difficulties in te numerical simulation of te scattering problem arise from te unbounded nature of te domain, and from te usually oscillatory nature of te solutions. Recent work as led to te development of algoritms tat are better able to andle te igly oscillatory nature of te solutions at ig frequency. Tese algoritms incorporate in teir trial and test spaces oscillatory functions suc as plane waves, Fourier Bessel functions, and products of low order polynomials wit plane waves in order to capture qualitative information about te oscillatory caracter of te solution. Tese metods include te Partition of Unity Metod PUM) of Melenk and Babuska,[] te Discontinuous Enricment Metod DEM) [], te Ultra Weak Variational Formulation UWVF) of Cessenat and Després [3]. Te UWVF as been applied to te Maxwell equations [4], linear elasticity [5], acoustic fluid-solid interaction [6] and to tin clamped plate problems [7]. More recently, te Plane Wave Discontinuous Galerkin PWDG) metod as been studied by Hiptmair et al. [8, 9, 0, ] as a generalization of te UWVF metod. In tis case te problem is cast in te form of a Discontinuous Galerkin metod, wit possibly mes dependent penalty parameters. Hiterto, since te focus as been to understand plane wave discretization, only approximate first order absorbing boundary conditions ave been considered in PWDG metods to truncate te exterior domain. However, unless te computational domain is large, first order absorbing boundary conditions result in large errors due to reflections from te artificial boundary. To reduce errors due to boundary reflections, witout taking te absorbing boundary far from te scatterer, several numerical tecniques ave been considered. Tese include ig order absorbing boundary conditions [], te Diriclet-to-eumann Dt) mapping [3, 4, 5], and perfectly matced layers [6]. Te Dt map can be enforced via boundary integral equations or Fourier series expansions resulting from te metod of separation of variables. In tis paper, we will couple te Fourier series Dt metod wit PWDG. In [4, 5], Koyama derives error estimates for te Dt finite element metod, considering bot te error due to truncation of te Dt series and due to finite element discretization. Hsiao et. al [3] derive error estimates for te conforming finite element Dt metod, and present numerical results tat sow optimal convergence in te L and H norms using conforming piecewise linear finite elements. Following [4, 5, 3], we consider an approximate boundary value problem wit a truncated Dt series. Our main contribution in tis work is to coose new numerical fluxes on te artificial boundary tat enforce te truncated Dt boundary condition, and tat are consistent wit respect to te solution of te truncated boundary value problem. In te P.D. tesis [7], A. Moiola derives approximation results for solutions of te Helmoltz equation by plane waves, using an impedance first order absorbing boundary condition. We modify tese arguments to take into account te non local Dt series, and to derive error estimates in tis case. Since te Dt map naturally maps from H on te artificial boundary, but plane wave basis functions are only in L, our error estimates deteriorate wit order as. Tis problem can be overcome by using te eumann-to-diriclet map td), and is part of ongoing work to incorporate generalized impedance boundary conditions into te PWDG metod.

4 Tis paper is organized as follows: In Section, we consider te continuous problems wit te full and truncated Dt series, and prove stability estimates in bot cases. In Section 3, we introduce te PWDG metod for solving te scattering problem, and sow te well-posedness and consistency of te PWDG sceme. In Section 4, we modify te error estimates in [7] to take into account te Dt boundary condition. Section 5, we describe te numerical implementation of te Dt map via a projection into te space of trigonometric polynomials. In Section 6, numerical results are presented to sow te effect of te wavenumber, truncation of te Dt series, number of plane waves, and mes size on te convergence of te numerical sceme. We end in Section 7 wit some concluding remarks.. Te Scattering Problem We consider te scattering of an acoustic wave from a sound-soft impenetrable obstacle occupying a region Ω D R. We assume tat te boundary of Ω D, denoted Γ D is smoot, and te exterior R \Ω D is connected. Te scattered field u satisfies te following Diriclet boundary value problem in R \Ω D : u + k u = 0 in R \Ω D a) u = g on Γ D := Ω D b) ) lim r u r r + u = 0, c) were r = x, x = x, x ), and g = u inc, were u inc is a known incident field. We will assume tat te wavenumber k > 0 is constant, altoug te algoritm applies to te more general cases were k is piecewise constant in eac element. Te well-posedness of te exterior Diriclet problem ) is demonstrated for example in Teorem 3. of [8]. To apply a domain based discretization, it is necessary to introduce an artificial domain Ω R wit boundary enclosing Ω D suc tat dist, Γ D ) > 0, and to introduce suitable boundary conditions on tat take into account wave propagation in te infinite exterior of Ω R. Te boundary value problem ) is ten replaced by a boundary value problem posed in te annulus bounded by on te outside and Γ D on te inside. To date, a crucial step taken in PWDG metods for te Helmoltz equation is to use te first order absorbing boundary condition u n + u = 0 on ) In tis paper, as in [9, 3], we are going to use te Diriclet-to-eumann map now in te context of te PWDG metod. 3

5 .. on-local Boundary Value Problem Te scattering problem ) is equivalent to te following boundary value problem see, e.g. Jonson and edelec [0]): u + k u = 0 in Ω, u = g on Γ D, u u = w, n = w n on Γ, R w + k w = 0 in R 3) \Ω, ) lim r w r r + w = 0. Here u, w are te scattered fields in te interior and exterior of Ω respectively, and g H ). If w is known on, te normal derivative n w can be computed by solving for w in R \D. Te Dt map S : H ) H ) is defined as S : w ΓR n w ΓR. If te truncating boundary is a circle, te map S can be written explicitly as a series involving Hankel functions. By using te polar coordinate system, separation of variables sows tat te general solution of te omogeneous Helmoltz equation w + k w = 0 in R \Ω is wr, θ) = [ ] α m H m ) kr) + β m H m ) kr) e imθ, 4) m Z were H m,) z) are Hankel functions of first and second kind, and of order m Z. Te Hankel functions are in turn defined by Bessel functions J m z) and eumann functions Y m z) H m,) z) = J m z) ± iy m z). For an introduction to Bessel and Hankel functions in te context of te Helmoltz equation, see Colton and Kress [8], or Cakoni and Colton []. Only te Hankel functions of te second kind are consistent wit te Sommerfeld radiation condition c), so only solutions of te form H m ) kr)e imθ represent outgoing waves. Tis implies tat te coefficients α m in te series expansion 4) vanis. If wr, θ) H ) is given, ten we can write w as a Fourier series, wr, θ) = m Z w m e imθ were te Fourier coefficients w m are given by w m R) = πr wr, ϕ)e imϕ dϕ. Tus, te solution w of te Helmoltz problem for r R is wr, θ) = m Z w m R) H) m kr) H m ) kr) eimθ. 5) 4

6 Taking te normal derivative of wr, θ), wic is simply te radial derivative w/ r we can write an explicit form of te Dt map S: SwR, θ) := w r R, θ) = k H) m kr) m Z H m ) kr) w mr) e imθ. 6) Using te Dt map, we may restrict te domain of problem 3) to Ω and te equations for u become u + k u = 0, in Ω u = g, on Γ D 7) u n Su = 0 on Γ. R Te boundary condition u/ n Su = 0 on is an exact representation of wave propagation in te exterior domain. Te solution u of te boundary value problem 7) is te restriction to Ω of te unique solution of te exterior scattering problem )... Stability of te adjoint problem Since we will later use duality arguments to derive error estimates, we consider te following adjoint problem to 7) u + k u = f, in Ω u = 0, on Γ D 8) u n S u = 0 on. were S is te L adjoint of te Dt map on and f L Ω) is suc tat suppf) Ω. Wavenumber dependent stability bounds for te scattering problem wit te impedance boundary condition u/ n u = g R on are sown in [9], Teorem.. In te following teorem, we state stability bounds for te scattering problem wit a Dt map on. We will assume tat te boundary of te scatterer is smoot, so tat u H Ω). Altoug not needed ere, te following result is of independent interest. Proposition.. Let u H Ω) be te analytical solution of te adjoint problem 8). Ten tere exist positive numbers C and C independent of u and f, but wose dependence on k and R is known suc tat u,ω + k u 0,Ω C f 0,Ω, 9) u,ω C f 0,Ω. 0) Proof: Te solution u of te adjoint problem 8) can be extended analytically by Hankel functions of te first kind to te exterior region R \Ω. Denote still by u tis analytic extension in te region R \D. Let Ω := B R 0)\D be te annulus bounded by te circle on te outside and by Γ D on te inside, were is a circle of radius R centered at te origin. Let ũ = χu were χ C 0 Ω), 0 χ, is a smoot cut-off function equal to one in a neigborood of Ω and zero in a neigborood of. Ten ũ satisfies ũ + k ũ = f in Ω, ũ = 0 on Γ D, n ũ + ũ = 0 on, 5

7 were f = { f in Ω χu) + k χu. in Ω\Ω By te first stability estimate in inequalities 3.5) of [8], tere exists some constant C independent of k, ũ and f suc tat ũ, Ω + k ũ 0, Ω CR f 0, Ω. ) Te product rule sows tat χu) = χ u + u χ + χ u. Hence f = χf + u χ + χ u were f is extended by zero to te exterior of Ω. Since we can coose we ave tat χ, χ C/R, χ C/R f 0, Ω C 0 + kr + ) ) k R k u, Ω + k u 0, Ω. By Lemma 3.5 of [9], te solution u of 8) satisfies te stability bound ) k u, Ω + k u 0, Ω + 4 ) kr f 0, Ω = + 4 ) kr Ten we ave f 0,Ω u,ω + k u 0,Ω ũ, Ω + k ũ 0, Ω C 0 R f 0, Ω were C f 0,Ω C := C 0 R + 4 ) kr + kr + ) k R for some constant C 0 tat is independent of R and k. To sow te stability result 0) recall from te second stability estimate 3.5) of [8] u, Ω C 0 + kr) f 0, Ω, were C is independent of k and u. Combining wit te results above gives u,ω C f 0,Ω were C := C 0 + kr)c. 6

8 .3. Some properties of Hankel functions We point out some properties of te Dt map tat will be needed to derive error estimates. Te result in Lemma. can be found, for example, in Lemma 3.3 []. Lemma.. For any w H ), it olds tat Im Sww ds 0. ) Te following lemma is identical to Lemma 3. of [3], but in contrast to te result in [3], we give te dependence of te constant on te wavenumber. Lemma.3. For any m Z\{0} and kr > 0, it olds tat For m = 0, we ave tat + m ) for some C independent of kr. H ) m kr) m H ) m kr) H ) 0 kr) H ) 0 kr) C H ) m kr) m + kr. 3) H ) m kr) Proof: We make use of te inequality.7) in [9] H ν ) ρ) ν ρ ) + ρ H ) ν ρ) 4ρ π 0 and inequality.4) in [9] ρ H ν ) + ), 4) kr ρ) π for any ν and ρ R suc tat ν and ρ > 0. To obtain inequality 3) for m, coose ν = m, ρ = kr. Wen m, te result follows from te identity H ) m ρ) = ) m H ) m ρ). To prove te inequality for m = 0, recall te asymptotic relations of Hankel functions wit small argument as ρ 0 see e.g. 0..6, 0.7., of [3]) { H ν ) i/π) ln ρ if ν = 0 ρ) i/π)γ)z 6) if ν =, and for large argument as ρ 5) H ν ) ρ) It follows from 6) and 7) tat H ) 0 ρ) H ) 0 ρ) = { /πρ)e iρ π/4) if ν = 0 /πρ)e iρ 3π/4) if ν =. ρ) 0 ρ) H ) H ) { C ρ ln ρ as ρ 0 as ρ. 7) 8) 7

9 Since /ρ ln ρ ) < /ρ for small enoug ρ, we ave by continuity of te Hankel functions in any bounded interval H ) 0 ρ) H ) 0 ρ) C + ). 9) ρ wen ρ > Truncated Boundary Value Problem In practical computations, one needs to truncate te infinite series of te Dt operator to obtain an approximate mapping written as a finite sum S w = m k H) m kr) H m ) kr) w me imθ for all w H ). Te boundary value problem 7) is replaced by te following modified boundary value problem wit a truncated Dt map: Find u H Ω) suc tat u + k u = 0 in Ω u = g on Γ D u n S u = 0 on. In Teorem 4.5 of [3] it is sown tat te truncated exterior eumann problem is well-posed for all sufficiently large. Following te same arguments, we can prove te following teorem for te truncated exterior Diriclet problem ). Teorem.4. Tere exists an integer 0 0 depending on k suc tat for any g H Γ D ) te truncated Diriclet boundary value problem ) as a unique solution, u H Ω) for 0. 0) ).5. Stability of te truncated adjoint problem For later use in te derivation of L norm estimates for te discretization error, we make use of duality arguments, wic in turn depend on te following adjoint problem to ): z + k z = f in Ω, z = 0 on Γ D, z n S z = 0, were f L Ω). Te following lemma gives a stability constant C 3 of te solution z in te H norm wit respect to te L norm of te data f. We note in particular tat C 3 is independent of. Tis first inequality in Lemma.5 below is proved in Lemma 4. of [4]. Te second inequality is proved in [0] before equation 9). Lemma.5. Assume 0 is large enoug so tat te truncated boundary value problem ) as a unique solution z H Ω). Ten tere exist positive numbers C 3 and C 4 independent of, z and f, but depending on k and R suc tat ) z,ω C 3 f 0,Ω 3) z L Ω) C 4 f 0,Ω. 4) 8

10 3. Te Plane Wave Discontinuous Galerkin Metod Let T denote a finite element partition of Ω into elements {K}. We sall assume tat all te elements K T are generalized triangles. A generalized triangle will be a true triangle in te interior of Ω but may ave one curvilinear edge if te triangle as a face on Γ D or. More general elements e.g. quadrilaterals, pentagons, etc) are possible. Te parameter represents te diameter of te largest element in T, so tat = max K were K is te diameter of te smallest K T circumscribed circle containing K. Denote by E te mes skeleton, i.e. te set of all edges of te mes, E I te set of interior edges, E D te set of edges on te boundary of te scatterer Γ D and E R te set of edges on te artificial boundary. As is standard in DG metods, we introduce te jumps and averages as follows. Let K +, K T be two elements saring a common edge e. Suppose n +, n are te outward pointing unit normal vectors on te boundaries K + and K respectively. Let v : Ω C be a smoot scalar valued piecewise defined function, and σ : Ω C a smoot vector valued piecewise defined function. Let x be a point on e. Define v + x ) := lim y x y K + vy). Te definitions of v, σ + and σ are similar. Te jumps are defined as and te averages are defined as [v ] := v + n + + v n, [σ ] := σ + n + + σ n, 5) {v } := v + + v ), {σ } := σ + + σ ). 6) Suppose u H Ω) is te exact solution of te omogeneous Helmoltz equation. In eac element K T, te weak formulation is: u v k uv ) dx u n K v ds = 0, 7) K were v is assumed piecewise smoot, and n K is te outward pointing unit normal vector on K. Tis smootness assumption allows us to take traces of v and v on K. Integrating equation 7) by parts once more leads to u v k v) dx + u v n K ds u n K v ds = 0. 8) K K K To proceed, we suppose te test function v belongs in te Trefftz space T T ) defined as follows: Let H s T ) be te broken Sobolev space on te mes Ten te Trefftz space T T ) is H s T ) := { v L Ω) : v K H s K) K T }. T T ) := { v L Ω) : v H T ) and v + k v = 0 in eac K T }. Because v + k v = 0 in K, equation 8) reduces to u v n K ds u n K v ds = 0. 9) K K 9 K

11 Te problem now is to find an approximation of u in a finite dimensional Trefftz subspace of T T ). Define te finite dimensional local solution space V pk K) of dimension p K on eac element K T : V pk K) := { w H K) : w + k w = 0 in K } and te global solution space } V T ) := {v L Ω) : v K V pk K) in eac K T were te local dimension p K can cange from element to element. Suppose tat in eac element K T, u is te unknown approximation of u in te local solution space V pk K) and σ := u is te flux. Ten, on K, we write u v n K ds σ n K v ds = 0, 30) K K for all v V pk K). At tis stage, u and σ are multi-valued on an edge e K K, since te trace from K could differ from tat of K. To find a global numerical solution in V T ), we need u and σ to be single valued on eac edge of te mes. Tus, we introduce numerical fluxes û and σ tat are single valued approximations of u and σ respectively on eac edge. In eac element of te mes K T, it olds tat û v n K ds σ n K v ds = 0. 3) K Integration by parts allows us to write a domain based equation tat is equivalent to 3) u v k u v ) dx + û u ) v n K σ n K v ds = 0. K K K Te form 3) is used to prove coercivity properties of te PWDG metod, wile te skeleton-based form 3) is used to program te metod. We now define te PWDG fluxes. Te definition of te fluxes on interior edges and edges on te scatterer are taken to be tose of standard PWDG metods in [9] and [8] as given in 33) and 34). But te fluxes on te artificial boundary are new. Denote by te elementwise application of te gradient operator. Following [9], û = {u } β [ u ], on interior edges E I. 33) σ = { u } α [u ] K 3) On te boundary of te scatterer Γ D } û = 0, σ = u αu on Diriclet edges E D. 34) For edges on te artificial boundary E R, we propose û = u δ u n S ) u, 35) σ = S u n δ S 0 u S u n ), 36)

12 were α, β, δ > 0 are positive flux coefficients defined on te edges of te mes, and S L )-adjoint of S, defined as S vw ds = vs w ds. is te Substituting tese fluxes into equation 3), and summing over all elements K T, we obtain te following PWDG sceme: Find u P W T ) suc tat for all v P W T ) A u, v ) = L v ) 37) were A u, v ) := u v k u v ) [[ ]] dx u { v } ds Ω E I S u v ds β [[ ]] u E R [ v ] ds E I {{ }} u [v ] ds + α [[ ]] u [v ] ds E I E I ) ) δ u n S u v n S v ds E R + αu v ds u v n + u nv ) ds, 38) E D E D and te rigt and side is L v ) := g v n ds + αgv ds. 39) E D E D Te ermitian sesquilinear form 38) allows us to prove coercivity of te Dt-PWDG sceme. However to program te metod, we can make te algoritm more efficient by exploiting te Trefftz property of P W T ) to write te sesquilinear form on te skeleton of te mes. Tis avoids te need to integrate over elements in te mes. Integrating by parts 3) and using te Trefftz property v + k v = 0, te elemental equation 3) reduces to û v n ds σ nv ds = 0. 40) K Ten substituting te numerical fluxes and summing over all elements of te mes, we get A u, v {{ }} {{ }} ) = u [ v ] ds u [v ] ds S u v ds E I E I E R + u v n ds β [[ ]] u E R [ v ] ds E I + α [[ ]] u [v ] ds u nv ds + αu v ds E I E D E D ) ) u n S u v n S v ds 4) E R δ K

13 Our Dt-PWDG MATLAB code is based on te sesquilinear form 4). For error estimates, it is useful to derive an equivalent form of 38). Applying a DG magic formula Lemma 6., [4]) to 38) we get A u, v [[ ]] [[ ]] ) = u {v } ds u { v } ds u v n ds E I E I E D β [[ ]] u [ v ] ds + α [[ ]] u [v ] ds E I E I ) + αu v ds+ u n S u v ds E D E R ) ) u n S u v n S v ds. 4) E R δ Proposition 3.. Te Dt-PWDG metod is consistent. Proof. If u is te exact solution of te truncated boundary value problem ), ten under te assumptions on te geometry of te scatterer, u H Ω), tus on any interior edge e, [u ] = 0 and [ u ] = 0 on E I, u n = S u on E R, and u = g on E D. Terefore from 4), for any v P W T ) A u, v) = g v n ds + E D αgv ds E D = L v). 43) Proposition 3.. Provided 0, te mes-dependent functional v DG := Im A v, v) 44) defines a norm on T T ). Moreover, setting v := v + k β {v } DG + DG 0,E + k α { v } I 0,E I +k α v n 0,E + k δ v 0,ER 45) D we ave A v, w) v DG 46) w DG + Proof. Taking te imaginary part of 38), we ave Im A v, v) = k β [ v ] 0,E + k α [v ] I 0,E + k α v 0,ED Im I S vv ds E R +k δ / v n S v) 0,E R = v. DG 47) From Lemma., we recall tat taking only partial sums, Im S vv ds = 4 v m E R H m ) kr) > 0 m

14 If Im A v, v) = 0, ten v H Ω) satisfies te Helmoltz equation v + k v = 0 in Ω, wit v = 0 on Γ D, and v n S v = 0 on our new coice of flux is key to asserting tis last equality). By Teorem.4, tis problem as only te trivial solution v = 0 provided 0 is large enoug. To prove 46), we apply te Caucy Scwarz inequality repeatedly to 4). Remark: Te assumption tat 0 < δ required to prove continuity of te sesquilinear form for te PWDG wit impedance boundary conditions as in [9, 8] is no longer necessary. It is sufficient tat δ > 0 in te Dt-PWDG sceme. Tis is because of our new coice of boundary fluxes. Proposition 3.3. Provided 0, te discrete problem 37) as a unique solution u P W T ). Proof. Assume A u, v) = 0 for all v P W T ). Ten in particular A u, u ) = 0 and so Im A u, u ) = 0. Ten u = 0 wic implies DG u = 0 since is a norm on P W T DG ). 4. Error Estimates 4.. A mes-dependent error estimate We state an error estimate in te mes dependent DG norm. Proposition 4.. Assume 0. Let u be te unique solution of te truncated boundary value problem ), and u P W T ) te unique solution of te discrete problem 37). Ten u u DG inf w P W T ) u w DG +. 48) Remark: Using Moiola s estimates for approximation by plane waves, tis result can be used to provide order estimates see Corollary 4.6 later in tis paper). Proof: Let w P W T ) be arbitrary. By Proposition 3., definition 44) and te inequality 46) we ave u u DG = Im A u u, u u ) A u u, u u ) = A u u, u w ) u u DG u w DG Error estimates in te L norm Let e = u u be te error of te Dt-PWDG metod, were u is te exact solution, and u te computed solution. Let u be te solution of te truncated boundary value problem ). Ten, by te triangle inequality u u L Ω) u u L Ω) + u u L Ω). 49) Te term u u L Ω) is te truncation error introduced by truncating te Dt map, wile te term u u L Ω) is te discretization error of te Dt-PWDG metod. 3

15 4... Estimation of u u L Ω) Error estimates for te truncation error of te Helmoltz problem 7) wit Dt boundary conditions were proved by D. Koyama see [5, 4]), were it is assumed tat suppf) Ω, Ω B a were B a is a ball of radius a R, suc tat B a B R. Te main result of tis section is te following see Proposition 4. of [4]). Teorem 4.. Assume 0 and tat te exact solution u of problem ) belongs to H m Ω). Assuming tat suppf) Ω, Ω B a were B a is a ball of radius a R, suc tat B a B R, tere exists a positive number C 4 independent of, f, u and u, but depending on k, a, R and Ω, suc tat for m {0} and s max{, m } we ave u u m,ω C 4 s+m H ) H ) kr) ka) R u; s, a) were Ω may be Ω and R u; s, a) := m > m s u m a) Estimation of u u L Ω) In tis section, we study a priori error estimates for te discretization error u u L Ω) of te Dt-PWDG sceme. For tis section, we make te following assumptions on te mes and flux parameters. Assumption on te mes: Te mes is sape regular and quasi-uniform. Te metod can be applied to more general meses suc as quadrilateral elements and locally refined meses, but our focus in tis capter is te boundary condition on, so we coose simple quasi-uniform meses. Assumption on te numerical fluxes: Since te mes is quasi-uniform, we assume te numerical fluxes α, β, δ are positive universal constants on te mes. Suppose e K K is a common edge between triangles K and K on boundary edges assume K is empty). We recall te trace inequality see [5], Teorem.6.6), v 0,e C j= K j v 0,K j + Kj v,k j ) 50) wit C depending only on te sape regularity measure µ. In addition, as v H Ω), it olds tat ) v 0,e C v K j 0,K j + Kj v,k j 5) j= wit C depending only on te sape regularity parameter µ. Coosing f = e := u u in ), we ave by te adjoint consistency of te Dt-PWDG sceme tat A w, z ) = we dx. 5) 4 Ω

16 were w T T ) is any piecewise solution of te Helmoltz equation. Coosing w = e in 5), te consistency of te Dt-PWDG metod in Proposition 3. implies tat e L Ω) = A e, z ) = A e, z z ) 53) for any arbitrary z P W T ). To approximate te rigt and side of 53) we follow te idea introduced in Lemma 5.3 of [6] and Lemma 3.0 of [7]: Let z c be te conforming piecewise linear finite element interpolant of z H Ω). Ten we can find a z P W T ) tat can approximate z c. Adding and subtracting zc, we ave e L Ω) = A e, z z c ) + A e, zc z ) For convenience of notation, denote by e,c := z z c te conforming error. It follows tat A e, e,c E ) = [[ ]] {{ }} [[ ]] {{ e e,c ds e e I EI,c )}} ds 54) E I β [[ e E R δ e n S e ]] [[ e,c )]] ds + ) e,c E I α [[ e n S e,c E D e e,c n ds+ E R e n S e ) e,c ]] [[ ]] e,c ds ) ds ds + αe e,c E D ds. As e,c is continuous, it follows tat [[ ]] e,c = 0 on eac interior edge, so tat α [[ e E I ]] [[ ]] e,c ds = 0. 55) Consider terms involving e,c in te sesquilinear form 54): [[ ]] {{ }} I = e e,c ds E I I = I 3 = e EI k β [[ e ]] 0,e k β e,c 0,e, αe ne,c ds E D α e 0,e k α e,c, 0,e e E D k e n S e E R e ER k δ ) e,c ds e n S e ) 0,e k δ e,c 0,e. 5

17 For any edge e E I, assume tat e K K. Under te assumption tat te numerical fluxes are constant, we ave by te trace inequality 50) tat β e,c 0,e C e,c l= 0,K l + Kl e,c 0,K l Kl C 3 Kl z,kl. 56) l= Te oter terms involving α, δ are estimated in exactly te same way. Terefore, I + I + I 3 Cµ, α, β, δ, k, R) e DG 3 K T K z,k 57) were C depends on te wavenumber k, radius R, sape regularity parameter µ and te flux parameters α, β and δ, but is independent of, z and. ow consider terms involving e,c n in 55). [[ ]] {{ }} J = e e,c ds E I J = J 3 = J 4 = e EI k α [[ e ]] 0,e k α β [[ ]] [[ ]] e e,c ds E I e EI {{ }} e,c 0,e, k β e 0,e k β e,c 0,e, e e,c n ds E D α e 0,e k α e,c, 0,e e E D k We start by approximating J. K, δ e n S ) e e,c n S ) e,c ds k δ e n S e ) 0,Γ k δ e,c R n S e,c ) 0,Γ. R α {{ }} e,c 0,e C By te trace inequality 5), we ave on an edge e K l= e,c Kl C Kl z,kl. l= 0,K l + Kl e,c,k l 6

18 Te same argument olds for J and J 3. Te term J 4 tat involves te Dt map can be estimated via te triangle inequality k δ e,c n S e,c ) 0, k δ e,c 0, + k δ S e,c 0,. 58) Te first term in 58) is estimated as before. ow we estimate te term wit te Dt map. For m Z, define m 0 as follows { m if m 0 m 0 := 59) if m = 0. Let T R := {K T : lengt K ) > 0} be te set of all elements wit an edge on. Ten te term in J 4 wit te Dt map is estimated as δ S z z c ) 0,Γ = πδ m 0 k R H m ) kr) R m 0 H m ) e,c kr) ) m m Cδ, k, R) e,c 0, Cδ, k, R) e ER e,c 0,e Cµ, δ, k, R) 3 K z,k. 60) K T R Hence, summarizing we ave te estimate J + J + J 3 + J 4 C + ) e DG z,k K T 6) Combining 57), 6) and.5), we arrive at A e, z z c ) C + ) z,ω e DG C + ) e DG u u 0,Ω. 6) were we ave used te quasi-uniformity of te mes and te stability estimate 3). To estimate te term A e, zc z ), we use results from Lemma 5.4 of [6]. By similar arguments used in te estimation of A e, z z c ), and using te second inequality of Lemma.5 to bound z L Ω), we get A e, zc z ) C 3 + ) e DG z L Ω) C 3 + ) u u L Ω) e DG C 3 + ) u u L Ω) e. DG ow, by combining 6) and 63), we ave proved te following teorem: 7

19 Teorem 4.3. Let u H Ω) be te solution of te truncated boundary value problem ) and u be te solution of te discrete problem 37). Ten tere exists a constant C depending only on Ω, te flux parameters α, β and δ and te sape regularity parameter µ, but independent of k,u, u,, and suc tat u u L Ω) C + ) inf w P W T ) u w DG +. 63) Proof: By 6) and 63) we ave u u L Ω) = A e, z z c ) + A e, zc z ) 64) Te result follows from te error estimate in te DG norm from Proposition 4.. We need to estimate te term inf w P W T ) u w DG + tat appears in Teorem 4.3 along te lines of results due to A. Moiola in [7] for te Helmoltz equation wit an impedance boundary condition. In [7], detailed error estimates for te approximation of solutions of te omogeneous Helmoltz equation by plane waves are sown. To derive tese, two assumptions are made. Te first concerns te domain K Ω in particular a generalized triangle) were we will approximate a solution of te Helmoltz equation by plane waves see Lemma 3.. of [7]). Te second assumption concerns te distribution of plane wave directions on te unit circle see Lemma [7]). ow assume w is a solution of te omogeneous Helmoltz equation and w H m+ K), were m Z. Assume also q m +. Define te k-weigted norm w l,k,k by w l,k,k = l k l j) w j,k j=0, w H l K), k > 0. 65) and for every 0 j m +, consider ε 0, ε, ε defined in equation 4.0) of [7]: ε j := [ logq + k) j+6) ) ] e ) m+ j) 4 ρ)k m+ j + + k)q m+, q c 0 q + )) q 66) were ρ is a parameter related to te sape regularity of te elements, suc tat for a mes wit sape regularity µ, ρ = µ). Te constant c 0 measures te distribution of te directions {d l } Lemma [7]). ow te general approximation result of Moiola is: Teorem 4.4. approximation by plane waves: Corollary 3.55 [7]) Let u H m+ K) be a solution of te omogeneous Helmoltz equation, were K R is a domain tat satisfies assumptions in Lemma 3.. of [7]. Suppose te directions {d l } l= q,,q satisfy te assumptions in Lemma of [7]. Ten tere exists α C p suc tat for every 0 j m +, u p α k e x d l j,k,k Cε j u m+,k,k 67) l= were C depends on j, m and te sape of K. 8

20 In te next Lemma, we state best approximation error estimates in te DG + norm for te Helmoltz equation wit a Dt boundary condition. Te error bounds for u ξ 0,E and u ξ) 0,E on te skeleton of te mes can be found in Lemma 4.4. of [7]. We state tem ere for completeness. For convenience of notation, define γ := R + kr) Lemma 4.5. Assume tat te directions {d l } satisfy te assumptions of Teorem 4.4. Given u T T ) H m+ Ω), m, q m +, tere exists ξ P W T ) suc tat we ave te following estimates u ξ Cε 0,E 0 ε0 ) + ε u 68) m+,k,ω u ξ) Cε E ε ) + ε u 69) m+,k,ω Γ Im S u ξ)u ξ) ds Ckε 0 ε 0 + ε ) u 70) m+,k,ω R were C is independent of,, k, p, ξ, {d l } and u. S u ξ) 0, Ck γε 0 ε0 + ε ) u m+,k,ω 7) u ξ DG + C [ ε 0 ε 0 + ε )k γ + k) + k ε ε + ε ) ] u m+,k,ω 7) Proof: We ave by te trace estimate and Teorem 4.4 ) u ξ C 0, K K u ξ + u ξ u ξ 0,K 0,K,K Cε 0 ε 0 + ε ) u, m+,k,ω ) u ξ) C 0, K K u ξ + u ξ u ξ,k,k,k Cε ε + ε ) u m+,k,ω. ow to prove te error estimate for terms wit te Dt map, recall Lemma. Γ Im S vv ds = 4 v m R H m ) kr) and m kr H m ) kr), m. π We ave, Γ Im S u ξ)u ξ) ds = 4 R m m u m ξ m H ) m kr) πkr u m ξ m Ck u ξ 0, Ck u ξ 0, K. K T R 9

21 Recalling te definition 59) of m 0, and by te bounds in 3) and 9) S u ξ) = πkr m H m ) kr) 0,Γ 0 R m 0 H m ) u m ξ m kr) To prove te last error estimate, note tat m C R + kr) u ξ 0, C R + kr) u ξ 0, K. K T R u ξ DG +, C k u ξ 0,E + k u ξ) 0,E + Γ Im S u ξ)u ξ) ds R +k S u ξ) 0, ). Te first term in te square brackets of 66) decays algebraically for large q wile te second term decays faster tan exponentially. Terefore for large q, we ave te following order estimates: Corollary 4.6. Given u T T ) H m+ Ω), m, q m +, large enoug suc tat te algebraic term in 66) dominates te exponentially decaying term, tere exists ξ P W T ) suc tat we ave te following estimate u ξ DG + C m + ) logq + ) q ) m u m+,k,ω Proof: Te result follows easily from Lemma 4.5 and te inequality u m+,k,ω C u m+,k,ω wic olds for some C independent of 0 see te proof of Teorem.5 in [4]). Using Corollary 4.6, we ave te following main teorem of tis section: Teorem 4.7. Let u be te solution of te truncated boundary value problem ) and u be te computed solution, m, q m +, large enoug suc tat te algebraic term in 66) dominates te exponentially decaying term. Tere exists a constant C tat depends on k and only as an increasing function of teir product k, but is independent of p, u, u and suc tat u u L Ω) ) logq + ) m C m + ) u m+,k,ω q. 73) Proof: Te result follows from Corollary 4.6 and Teorem 4.. Remark: If k,, q, R are fixed in te error estimate 73), ten were C is independent of. u u C m u m+,k,ω 0

22 5. umerical Implementation Let = dimp W p T )) be te total number of degrees of freedom associated wit te PWDG space P W T ). Obviously = K T p K. Te algebraic linear system associated wit te Dt-PWDG sceme is AU = F 74) were A C is te matrix associated wit te sesquilinear form A, ) 4), and F C is te vector associated wit te linear functional L ), and U C, te vector of unknown coefficients of te plane waves in P W T ). More precisely, te discrete solution can be written in terms of plane waves u = p K K T l= u K l ξk l 75) were te coefficients u K l C, and te basis functions ξk l are propagating plane waves { ξ K l = expx d K l ) if x K 0 elsewere and te directions are given by ) )) πl πl d K l cos =, sin p K p K Rewriting 75) in vector form u = u j ξj. j= ten U = [ u,, u ] T. Te global stiffness matrix associated wit te sesquilinear form A u, v ) is A = A Int + A Dir + A R,loc + A Dt were A Int is te contribution from interior edges E I, A Dir is te contribution from edges on te Diriclet boundary E D. Terms in A u, v ) defined on E R but wit no Dt map contribute A R,loc. Teir computation is standard for PWDG metods. However te term A Dt is computed globally since te Dt map involves an integral on te entire boundary, and we give details of te calculation now. of trigonometric polynomi- To compute te stiffness matrix A Dt, we introduce te space W als { } := span e inθ : n. W

23 Te projection operator P : L ) W on te artificial boundary onto te + dimensional space of trigonometric polynomials is defined as P ϕ ϕ) η ds = 0, 76) were ϕ L ), and η W. In practice, integrals on Legendre quadrature using 0 points per edge. Let w be te projection of te computed solution, u : are computed elementwise by Gauss- w := P u = were η l = e ilθ is a basis function of W. Ten w l = πr u j j= l= w l η l, ξ j η l ds = πr M lj u j j= were te components of te projection matrix M C +) M lj = ξj η l ds. are defined as Wit tese coefficients, we can write W = πr MU, were U is te vector of te unknown coefficients of u Let v = ξ j. Ten W = [ w,, w ] T. P v = P ξ j = πr Me j previously defined, and were e j is a vector wit one on te j t coordinate, and zero oterwise. Ten coosing v = ξj, te Dt term is computed as S u v ds := SP u )P v ) ds = πr M T MU) j, 77) j =,, and M is te conjugate transpose of M, and T is te + ) + ) diagonal matrix ζ ζ T = ζ

24 wit diagonal entries ζ m = k H) m kr). H m ) kr) To compute terms on involving normal derivatives, observe tat u n = u j d j n)ξj. 78) j= Let We ave Ten were πrz l = z = P u n) = = P u j j= m= z l η l. u j d j n)ξj η l ds j= d j n)ξ j η l ds. 79) Z = πr M D U, Z = [ z,, z ] T and M D is te + ) projection-differentiation matrix wit entries M D ) lj = d j n)ξj η l ds. Coosing v = ξ, we ave j δ u n S u ) ξj n S ξ j ) ds = δ u n) ξ j n) ds + δs u ) ξj + δ u n)s ξ j ) ds δs u )S ξ j ) ds n) ds = I + I + I 3 + I 4. 80) Te first term I is computed locally on eac edge, in te standard way. Te second term I is computed as follows I = δ ) ) SP u ξ n ds j = δ πr [M D T MU] j. 8) 3

25 Te tird term is computed as To compute te fourt term, I 3 = δ ΓR ) u n) SP ξj = ds δ πr [T M) M D U] j. 8) I 4 = δsp u )SP ξ j ) ds = δ πr [T M) T M)U] j. 83) It follows tat te contribution to te global stiffness matrix from terms involving te Dt map on is te matrix A Dt = πr M T M + δ [ M D πr T M + T M) M D T M) T M) ]. 84) 6. umerical Results In tis section, we numerically investigate convergence of te Dt-PWDG sceme. We consider te scattering of acoustic waves by a sound-soft obstacle, as modeled by te boundary value problem ). In our numerical experiments, we consider bot te impedance boundary condition IP-PWDG) for te scattered field and te Dt boundary condition, u n + u = 0, on 85) u n S u = 0, on. 86) In all numerical experiments, te Diriclet boundary condition is imposed on te scatterer u = u inc, on Γ D. 87) were u inc = ex d is a plane wave incident field propagating in te direction d relative to a negative orientation of te xy axis, due to our coice of te time convention in te definition of te time-armonic field. Te computational domain is te annular region between and Γ D. For scattering from a disk, te scatterer Ω D is a circle of radius a = 0.5, wile te artificial boundary on wic te Dt map is imposed is a circle of radius R =, centered at te origin. All computations are done in MATLAB. Te code used in our numerical experiments is based on te D Finite Element toolbox LEHRFEM [8]. Te outer edges on r = a, r = R are parametrized in polar coordinates, and ig order Gauss-Legendre quadrature 0 points per edge) is used to compute all integrals defined on edges on and Γ D. Curved edges are used in order to eliminate errors tat arise from using an approximate polygonal domain. 4

26 Te exact solution of te scattering problem ), in te case of a circular scatterer, is given in polar coordinates by see Section 6.4 of Colton [9]) [ J 0 ka) ur, θ) = ka)h) ] H ) 0 kr) + i m J mka) 0 m= H m ) ka) H) m kr)cos mθ) 88) For numerical experiments, we take = 00 to truncate te exact solution. Tis value of is sufficient for te wavenumbers considered. Experiment : Scattering from a disk Our main example is a detailed investigation of te problem of scattering of a plane wave from a disk. We coose te Dt or impedance boundary to be concentric tis improves te accuracy of te impedance boundary condition). In Fig, we compare density plots of te approximate solution by bot metods using te same discrete PWDG space. Comparison of te sadow region of te impedance boundary condition solution in Fig top left) wit tat of te exact solution demonstrates te effect of spurious reflections from te artificial boundary. Te Dt-PWDG solution in te top rigt panel of Fig sows greater fidelity wit te exact solution. Tere is little difference between te sadow regions of te Dt solution and te exact solution. It is interesting to note tat from te plots of te real parts of te solution in te bottom row of Fig, tere is little obvious difference between te solutions. However our upcoming and more detailed analysis sows significant improvements from te Dt boundary condition. Our first detailed study investigates te error due to truncation for te Dt-PWDG. We fix a grid and PWDG space and vary for several wavenumbers k. Results are sown in Fig. Results from te top left panel of Fig suggest tat for all values of k considered tere exists an 0,k suc tat no furter improvement in accuracy is possible for > 0,k, for a fixed number of plane waves p and a fixed mes widt. Taking > 0,k does not improve te accuracy of te solution. Te error is ten due to te PWDG solution. Tere are tree pases in te plots: i) a pre-convergence pase, were increasing as little effect on convergence ii) convergence pase were rapid exponential convergence of te relative error wit respect to is observed. iii) postconvergence pase wen > 0,k and optimal convergence as been reaced. Te top rigt plot suggests tat 0,k. kr is sufficient to reac te optimal order of accuracy, wic agrees wit te roug numerical rule of tumb 0,k > kr. In te bottom plot, we note tat te exponential rate of convergence in te convergence pase is independent of, owever te final 0,k and error depend on. ext, we fix = 30 sufficiently large on te basis of te previous numerical results tat te error due to te truncation of te Dt map is negligible) and examine -convergence for te Dt-PWDG and standard PWDG wit an impedance boundary condition. Results are sown in Fig 3. Te left grap suggests tat te rate convergence of te relative error wit respect to is independent of k since all curves are rougly parallel. For all k considered, te rate of convergence for p = 7 is rougly te rate of 3.5 wic exceeds te rate of about 3.0 predicted in Teorem 4.7. Te actual relative error owever, depends on k, as expected from consideration of dispersion: iger values of k result in more dispersion error wic is unavoidable even in te PWDG metod []. Te rigt plot is for PWDG wit an impedance boundary condition. It suggests limited convergence of te relative error computed using impedance boundary conditions. However, te accuracy of 5

27 Figure : Scattering from a sound-soft disk: a = 0.5, R =, p = 5 plane waves per element, k = 7π, = 0 Hankel functions. Top left: absolute value of te solution computed using impedance boundary conditions. Top rigt: absolute value of te solution using Dt boundary conditions. Middle left: te mes. Middle rigt: absolute value of te exact solution. Bottom left: real part computed using te impedance boundary conditions. Bottom rigt: real part computed using te Dt boundary conditions. 6

28 Figure : Scattering from a disk: Top left: semi-log plot of te relative L -norm error vs maximum order number of te Hankel functions in te Dt expansion, for k = 4, 8, 6, 3, p = 7, = 0.. Top rigt: log of te relative L -norm error vs /kr, p = 7, = 0.. Bottom: log of te relative L -norm error vs, for k = 8, p =, = /5. Figure 3: Scattering from a disk: log-log plot of te relative L -norm error vs /. Left: Dt- PWDG wit = 30, p = 7 plane waves per element. Rigt: IP-PWDG, p = 7. 7

29 Figure 4: Scattering from a disk: Left: log-log plot of te relative L error vs /. Rigt: empirical rates of -convergence for different values of p. Figure 5: Scattering from a disk: log of te relative L error vs p te number of plane waves per element. Left: impedance boundary condition. Rigt: Dt boundary condition wit = 30, = 0.. te solution increases wit te wavenumber, in a way contrasting wit te results of te top grap for Dt-PWDG. Tis suggests tat te errors due to te approximate boundary condition exceed tose due to numerical dispersion in tis example. Our next example examines convergence for different coices of p. We only consider te Dt- PWDG because of te adverse error caracteristics te impedance PWDG sown in Fig 3. Results for te and p study are sown in Fig 4. Te results in Fig 4 left panel sow te increased rate of convergence of te PWDG wen p is increased. Tis is clarified in te rigt panel. As expected from Teorem 4.7, increasing te number of directions of te plane waves per element results in a progressively iger order sceme. Our final numerical study for scattering from a disk examines p convergence of te Dt-PWDG and impedance PWDG. Results are sown in Fig 5 were we fix and te mes size. From Fig 5, as in te case of -convergence, te impedance boundary condition sows limited 8

30 convergence up to a relative error of about 0%, again suggesting tat te error due to te boundary condition dominates te error due to te PWDG metod in tis example. Te relative L error for te Dt boundary condition converges exponentially fast wit respect to p, te number of plane wave directions per element. However convergence stops due to numerical instability caused by illconditioning at a relative error of 0 4 %. From tese experiments we note tat te critical number of plane waves needed before numerical instability sets in depends on te wavenumber. Experiment : Scattering from a resonant cavity In our next experiment in Fig 6 we sow results for a resonant L-saped cavity. Tis domain does not satisfy te geometric constraint tat te scatterer is star-saped wit respect to te origin. Te domain can be included in our teory except we can no longer state k-dependent continuity and error estimates. Te solution will still be in H 3 +s Ω) for some s > 0. We consider scattering of a plane wave e x d from a non-convex domain wit an L-saped cavity in te interior, were te direction of propagation of te plane wave is d = ). Te top left and rigt panels of Fig 6 sow te absolute value of te scattered field computed using IP-PWDG and Dt-PWDG respectively. Te IP-PWDG results sow reflections on te rigt and side of te domain Ω reminiscent of te poor results in te sadow region for scattering from te disk. Tese reflections are not visible in te sadow region of te Dt-PWDG solution. Experiment 3: Scattering from a disconnected domain In te final set of experiments, we consider scattering of a plane wave incident field from a disconnected domain. Te top left solution in Fig 7 computed using IP-PWDG sows te effect of spurious reflections compared wit te smooter sadow region in te Dt-PWDG solution reminiscent of te results in Experiment for scattering from a disk. Te real parts of te scattered field are identical to te eye. Tis experiment demonstrates te importance of an improved treatment of te absorbing boundary condition for disconnected scatterers. 7. Conclusion We ave sown ow to couple te PWDG sceme to a non-local boundary condition in order to provide accurate truncation for te scattering problem. Bot convergence analysis and numerical results demonstrate good accuracy of te metod. Future work would be to use boundary integral equations to approximate te Dt equation. Our formulation sould generalize to tat case. 8. Acknowledgements Te work of te autors was supported in part by SF grant number DMS-660 and AFOSR Grant number FA Te autors acknowledge te support of te Institute for Matematics and its Applications, University of Minnesota during te special year Matematics and Optics wit funds provided by te ational Science Foundation. 9

31 Figure 6: Scattering from a domain wit an L-saped cavity, p = 5 plane waves per element, k = 5π. Top left: absolute value of te scattered field, IP-PWDG. Top rigt: absolute value of te scattered field, Dt-PWDG. Middle left: real part of te scattered field, IP-PWDG. Middle rigt: real part of te scattered field, Dt-PWDG. Bottom left: Te incident field in te direction d = ). Bottom rigt: mes 30

32 Figure 7: Scattering from a disconnected domain: k = π, p = 5 plane waves per element. Top left: absolute value of te scattered field, IP-PWDG. Top rigt: absolute value of te scattered field, Dt-PWDG. Middle left: real part of te scattered field, IP-PWDG. Middle rigt: real part of te scattered field, Dt-PWDG. Bottom left: incident field in te direction d = 0). Bottom rigt: mes. 3

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