Hybrid (DG) Methods for the Helmholtz Equation

Transcription

1 Hybrid (DG) Methods for the Helmholtz Equation Joachim Schöberl Computational Mathematics in Engineering Institute for Analysis and Scientific Computing Vienna University of Technology Contributions by A. Sinwel (Pechstein), P. Monk M. Huber, A. Hannukainen, G. Kitzler RICAM Workshop, Nov 2011, Linz Joachim Schöberl Page 1

2 Helmholtz Equation Heterogeneous Helmholtz Equation u + n(x)2ω 2u = f Impedance trace boundary condition u + iωu = iωgin n for convenience. better: PML, pole-condition,... Weak form: Find u H 1 s.t. Z u v n2ω 2uv + iω Ω Joachim Scho berl Z Z uv = Ω gv v H1 Ω Page 2

3 Weak and dual formulation Mixed formulation. Introduce velocity σ := 1 iω u: u iωσ = 0 div σ iωn 2 u = 1 iω f σ n + nu = g in Dual formulation. Eliminate u: 1 n 2 div σ + ω2 σ = 1 n 2f Weak form: i ω 1 div σ div τ iω n2 στ + Ω 1 n σ nτ n = f div τ + Ω g in τ n Joachim Schöberl Page 3

4 Function space H(div) Dual formulation requires the function space H(div) = {τ [L 2 ] d : div τ L 2 } has well-defined normal-trace operator Raviart-Thomas Finite Elements: V RT k = { a + b x : a [P k ] d, b P k } There holds [P k ] d V RT k [P k+1 ] d, div V RT k = P k, tr n,f V RT k P k (F ) Degrees of freedom are moments of normal traces on facets F of order k moments on the element T of order k 1 dofs ensure continuity of normal-trace Joachim Schöberl Page 4

5 Hybridization of RT -elements Arnold-Brezzi hybridization technique: Do not incorporate normal-continuity σ T1 n 1 = σ T2 n 2 into the finite element space, but enforce it by additional equations: F [σ n ]v F = 0 v P k (F ) Hybrid system: Find σ V dc RT k, u F P k (F ) such that T i ω u F is the primal variable on the facet. T div σ div τ iω T στ + T T τ nu F = 0 τ T T σ nv F + Ω uf v V = Ω g inv F v F Wish to eliminate σ from first equation (small local problems!), BUT local Dirichlet - problems are unstable for hω 1 Joachim Schöberl Page 5

6 Hybridization of Helmholtz equation Add a second Lagrange multiplier [Monk+Sinwel+JS 11] σ F n F := σ F n F and stabilize by adding 0 = T (σ n σ F n )(τ n τ F n ) F u σ σ F n σ T i ω T div σ div τ iω T στ + T σ nτ n + T T σ nv F σ n τ F n + T T T τ nu F τ n σ F n = 0 τ T σf n τ F n + Ω uf v F = Ω g inv F v F The element-variable σ can now be eliminated from the stable local equation (Robin - b.c.): i ω T div σ div v iω στ + T σ n τ n = T (σ F n u F )τ n Joachim Schöberl Page 6

7 Analysis Unique discrete solution and convergence rate estimates by usual duality technique (h sufficiently small) Qualitative property: Impedance trace isometry Local solvability (by Melenk test-functions τ = x div σ) Joachim Schöberl Page 7

8 Impedance trace isometry Element equation reads as i ω T div σ div v iω στ + T σ n τ n = T (σn F u }{{ F )τ } n g in define element-wise out-going impedance trace g out P k (F ) via i ω T div σ div v iω στ T σ n τ n = T g out τ n Choose τ = σ and take real parts: σ n 2 T = R{ } g in σ n So we get g out 2 T = g in 2σ n 2 T = g in 2 4R g in σ n + 4 g in 2 = g in 2 T The facet-equations can be rephrased as (for neighbour elements T ). g in = g out (σ T ) Joachim Schöberl Page 8

9 Convergence plots Comparing Hybrid-Helmholtz solution with conforming H 1 -FEM. σ RT k, σ F n, u F P k (F ). Postprocessing to ũ P k+1 (T ) L-2 error u RT3 - postproc std FEM err e-05 1e-06 1e-07 1e h Hybrid-Helmholtz has about twice the global dofs of standard FEM. Joachim Schöberl Page 9

10 How to solve???? Want to iterate for facet-variables only. Idea: In the wave-regime the wave-relaxation g in := g out (σ ˆT ) converges well. But, does not work for the elliptic regime when hω is small. Try to combine with an elliptic preconditioner. Joachim Schöberl Page 10

11 Facet dofs Facet - dofs are Dirichlet and Neumann data u F, σ F n, or left- and right-going traces g l, g r transformation ( g l g r ) = ( ) ( u F σ F n ) holds point-wise since all variables are in P k (F ) Joachim Schöberl Page 11

12 Motivation: 1D system The 1D system (with exact local problems) reads as ( 0 γ 0 0 ) ( g l j 1 g r j 1 ) + ( ) ( g l j g r j ) + ( 0 0 γ 0 ) ( g l j+1 g r j+1, ) left and right-going plane waves with phase-shift γ. left (right) going Gauss-Seidel for g l (g r ) is an exact solve. Matrix is not diagonal dominant. But, exchanging rows pairwise gives a diagonal-dominant, non-symmetric matrix. Krylov-space methods with most diagonal-like preconditioners will converge. Joachim Schöberl Page 12

13 Balancing domain decomposition methods Domain decomposition, Ω = Ω i, distribute elements Sub-assemble fe matrices with sub-domain matrix A i A = A i Balancing DD Preconditioner: Â 1 = R i A 1 i Ri T Elliptic case: local Neumann problems are singular! Solution: BDD with constraints BDDC (Dohrmann, Widlund,...) R A cc A c1 A cm A 1c. A A mc A mm 1 R T Analysis for the elliptic case with log α condition numbers. Joachim Schöberl Page 13

14 Heuristics: BDDC for the Hybrid Helmholtz system Bilinearform B(σ, σ F, u F ; τ, τ F, v F ) = i 1 div σ div τ iω στ + σ n v F + τ n u F + ω T T + (σ n σn F )(τ n τn F ) γ σn F v F + τn F u F T T The new blue term is consistent and is motivated by the g l g r row-exchange. Apply BDDC preconditioner with mean-value facet-dof constraints Good values for γ are found by experiments. Joachim Schöberl Page 14

15 Iteration numbers with coarse grid, 4 domains L/λ p= p= p= p= no coarse grid, 4 domains L/λ p= p= p= p= with coarse grid, 16 domains L/λ p= p= p= p= no coarse grid, 16 domains L/λ p= p= p= p= Joachim Schöberl Page 15

16 Infrastructure general purpose hp - Finite Element Code Netgen/NGSolve C++, open source MPI - parallel Dell R 910 Server 4 Xeon E7 CPUs GHz 512 GB RAM Joachim Schöberl Page 16

17 Diffraction from a grating Sphere with D = 40λ, 127k elements, h λ, p = 5, 39M dofs (corresponds to 9.4M primal dofs) 78 sub-domains / processes, no coarse grid T ass = 9m, T pre = 12m, T solve = 21m, 156 its Joachim Schöberl Page 17

18 Computational Optics factor out wavy part lense, n = 2 u = eik xu simple guess: k = n ω 0 better: ray-tracing, Eukonal equation smooth u Joachim Scho berl wavy u Page 18