The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B

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2Contrary to stiffness matrices arising from the finite element discretization, matricesgenerated by the discretized integral operators are dense and the memory and computationalrequirements can be high. However, new fast boundary element methods have beendeveloped to reduce this drawback (for the ACA method see, e.g., [17]).Because the Laplace equation can be considered as a special case of the Helmholtzequation, this work is related to the bachelor thesis [19] and the paper [20] describingthe boundary element method for the Laplace equation in 2D. This thesis is divided intoseveral sections, together building a scheme for the application of the boundary elementmethod for solving the Helmholtz equation in 3D. In the first section we introduce functionspaces necessary for the boundary integral formulation of the problem. In the followingpart we show the connection between the Helmholtz equation and the well-known waveequation and provide the representation formulae for the solution. In the third section weintroduce boundary integral operators and describe their properties. Afterwards, we deriveboundary integral equations used for the computation of the missing Cauchy data. In thelast part of this thesis we describe the discretization of the boundary integral equationsand provide some thoughts useful for practical implementation. Lastly, we provide somenumerical experiments.
31 Function SpacesIn this section we introduce function spaces necessary for the boundary integral formulationof boundary value problems for the Helmholtz equation. For a more detailed treatment ofthis topic we refer to [1] and [11].For d ∈ N we define a multiindexα = [α 1 , . . . , α d ] ∈ N d 0of the length |α| := α 1 + · · · + α d .u: R d → R can thus be expressed asPartial derivatives of a smooth enough functionD α u :=∂ |α|∂x α 11 . . . u.∂xα ddFor a complex-valued function u: R d → C we define the partial derivatives asD α u := D α (Re u) + iD α (Im u).1.1 Continuous FunctionsIn the following text we assume that Ω denotes a domain, i.e., a non-empty open connectedsubset of R d . By C(Ω) = C 0 (Ω) we denote the space of functions defined and continuousin Ω. For k ∈ N we denote by C k (Ω) the space of k times differentiable functions suchthat∀α, |α| ≤ k : D α u ∈ C(Ω).We also defineC ∞ (Ω) := k∈NC k (Ω).Moreover, for k ∈ N 0 ∪ {∞} we define the space C0 k (Ω) asu ∈ C0 k (Ω) ⇔ u ∈ C k (Ω) ∧ supp u := {x ∈ Ω : u(x) ≠ 0} ⊂ Ω is compact .By C(Ω) = C 0 (Ω) we denote the space of functions in C(Ω) that are bounded anduniformly continuous in Ω. Note that such functions are continuously extendable to ∂Ωasu(x) :=lim u(˜x)Ω∋˜x→x∈∂Ωfor x ∈ ∂Ωand in the following text we assume that these functions are already extended in sucha way. Similarly as above, C k (Ω) with k ∈ N represents the space of all functions inu ∈ C k (Ω) such that D α u ∈ C(Ω) for all α, |α| ≤ k. Equipped with the norm∥u∥ C k (Ω) := sup |D α u(x)|x∈Ω|α|≤k
Page 3: I hereby declare that this thesis i
Page 7: AbstractIn this work we study the a
Page 13: 1IntroductionThe boundary element m
Page 17 and 18: 5∂Ωy i 2ΩU + iU − iΓ iy i 1F
Page 19 and 20: 7For a special choice of p = 2 we g
Page 21 and 22: 9Note that the trace operator is no
Page 23: 11Definition 1.8 (Weak Solution). C
Page 27: 152 Helmholtz EquationLet us first
Page 32 and 33: 20 2 Helmholtz Equation∂ΩΩ εε
Page 34 and 35: 22 2 Helmholtz Equationthe normal v
Page 36 and 37: 24 2 Helmholtz Equationand due to p
Page 38 and 39: 26 2 Helmholtz Equationand thus l
Page 41 and 42: 293 Boundary Integral EquationsAt t
Page 43 and 44: 313.1 Single Layer Potential Operat
Page 45 and 46: 33For the jump of the Neumann trace
Page 47 and 48: 35with the surface curl operatorcur
Page 49 and 50: 37The following theorems are standa
Page 51 and 52: 39u ∈ H 1 (Ω, ∆+κ 2 ) if and
Page 53 and 54: 41Another approach to computing the
Page 55 and 56: 433.5.3 Interior Mixed Boundary Val
Page 57 and 58: 45respectively.The unique solvabili
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52 4 Discretization and Numerical R
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755 Numerical ExperimentsIn this pa
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77E N Err N Err N,p Err ϑ3962 1983
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792002000.81500.81500.60.41000.60.4
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81The approximate solution u h to (
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83with the unit direction vector d
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8540.540.430.430.220.320100.20.110
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87ConclusionIn the preceding sectio
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90[15] SAUTER, Stefan; SCHWAB, Chri
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