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

More documents

Recommendations

Info

50 4 Discretization and Numerical Realizationξ 2x k 31x k 1τ kˆτ(a) Reference triangle.1ξ 1x 3x 2x k 2x 1(b) General triangle.Figure 4.1: Triangular mesh elements.ϕ k (x)ψ k (x)x kτ k(a) Piecewise constant basis function.(b) Piecewise affine basis function.Figure 4.2: Basis functions.Furthermore, we define the linear space T ψ (∂Ω) asT ψ (∂Ω) := span{ψ k } E k=1 ,where E denotes the number of elements. Obviously, dim T ψ (∂Ω) = E and every complexvaluedfunction g ψ ∈ T ψ (∂Ω) can be represented asg ψ =Eg k ψ kand thus it can be identified with a vector g := [g 1 , . . . , g E ] T ∈ C E .k=1A smooth enough function g can be approximated by a linear combination of piecewiseconstant functions in several ways. The simplest approach is to setg k = g(x k∗ )
51with x k∗denoting the midpoint of τ k . Alternatively, we can setg k = 1 3g(x k 1) + g(x k 2) + g(x k 3) .However, since we deal with functions in Lebesgue and Sobolev spaces, where we do notdistinguish between functions differing on a measure zero set, the above mentioned approximationsmay prove impossible. Hence, we introduce the projection P ψ : L 2 (∂Ω) → T ψ (∂Ω)minimizing the error∥g − P ψ g∥ L 2 (∂Ω),i.e.,P ψ g := arg min ∥g − g ψ∥ Lg ψ ∈T ψ (∂Ω) 2 (∂Ω).To find the coefficients g kf : R E → R as E f(g 1 , . . . , g E ) := g − 2g k ψ kk=1= ⟨g, g⟩ L 2 (∂Ω) − 2For the partial derivatives of f we getfor a real-valued function g we introduce the functionL 2 (∂Ω)=g −Eg k ψ k , g −k=1Eg k ⟨g, ψ k ⟩ L 2 (∂Ω) +k=1∂f∂g j= −2⟨g, ψ j ⟩ L 2 (∂Ω) + 2Moreover, for the Hessian of f we haveH f [j, i] =EEg k ψ kk=1Eg kk=1 l=1Eg k ⟨ψ k , ψ j ⟩ L 2 (∂Ω).k=1∂2 f∂g j ∂g i= 2⟨ψ i , ψ j ⟩,L 2 (∂Ω)g l ⟨ψ k , ψ l ⟩ L 2 (∂Ω).which is the double Gram matrix corresponding to the basis of T ψ (∂Ω). Hence, the Hessianis positive definite and to find the minimum of f we set ∂f∂g j= 0 to obtain the system oflinear equationsEg k ⟨ψ k , ψ j ⟩ L 2 (∂Ω) = ⟨g, ψ j ⟩ L 2 (∂Ω) for j ∈ {1, . . . , E}k=1with a unique solution g = [g 1 , . . . , g E ] T ∈ R E . Because∆ k for k = j,⟨ψ k , ψ j ⟩ L 2 (∂Ω) =0 otherwise,
Page 3:
I hereby declare that this thesis i
Page 7:
AbstractIn this work we study the a
Page 13 and 14: 1IntroductionThe boundary element m
Page 15 and 16: 31 Function SpacesIn this section w
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
Page 59: 47with an unbounded domain Ω ext :
Page 64 and 65: 52 4 Discretization and Numerical R
Page 87 and 88: 755 Numerical ExperimentsIn this pa
Page 89 and 90: 77E N Err N Err N,p Err ϑ3962 1983
Page 91 and 92: 792002000.81500.81500.60.41000.60.4
Page 93 and 94: 81The approximate solution u h to (
Page 95 and 96: 83with the unit direction vector d
Page 97 and 98: 8540.540.430.430.220.320100.20.110
Page 99: 87ConclusionIn the preceding sectio
Page 102: 90[15] SAUTER, Stefan; SCHWAB, Chri
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?