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

76 5 Numerical Experimentsand the matrix formulationV κ,h g N = 12 M h + K κ,hg D .The approximate solution u h to (3.24) is given by the discretized representation formula(4.11).E N Err N Err N,p Err ϑ320 162 1.78 · 10 −1 1.24 · 10 −1 1.61 · 10 −31240 622 9.68 · 10 −2 6.48 · 10 −2 1.23 · 10 −47432 3718 3.33 · 10 −2 2.62 · 10 −2 5.44 · 10 −6Table 5.1: Interior Dirichlet BVP on the sphere.E N Err N Err N,p Err ϑ300 152 3.10 · 10 −1 1.81 · 10 −1 3.41 · 10 −31200 602 1.25 · 10 −1 9.10 · 10 −2 2.40 · 10 −47500 3752 4.22 · 10 −2 3.65 · 10 −2 9.55 · 10 −6Table 5.2: Interior Dirichlet BVP on the cube.10.820015010.82001500.60.41000.60.4100500.20−0.25000.20−0.20−50−0.4−50−0.4−100−0.6−0.8−100−0.6−0.8−150−1−1−0.500.500.51 1(a) Real part.−0.5−1−150−200−1−1−0.500.500.51 1(b) Imaginary part.−0.5−1−200−250Figure 5.2: Solution to the interior Dirichlet BVP on the sphere with E = 7432.In Tables 5.1, 5.2, 5.3 we summarize the results. The first two columns correspondto the mesh properties, namely to the number of elements and nodes. In the subsequentcolumns we provide L 2 relative errors given byErr N := ∥g N − g N,h ∥ L 2 (∂Ω)∥g N ∥ L 2 (∂Ω), Err N,p := ∥g N − g N,p ∥ L 2 (∂Ω)∥g N ∥ L 2 (∂Ω), Err ϑ := ∥u − u h∥ L 2 (ϑ)∥u∥ L 2 (ϑ)(5.3)