Preconditioners for the p-Version of the Boundary Element Galerkin Method in IR3 Vom Fachbereich Mathematik und Informatik der Universit¨atHannover zur Erlangung der venia legendi f¨urdas Fachgebiet Mathematik genehmigte Habilitationsschrift von Norbert Heuer aus Hannover Referenten: Prof. Dr. E. P. Stephan Prof. Dr. U. Langer Prof. Dr. M. Suri Habilitationsgesuch 4. Mai 1998 Habilitationsvortrag 17. Dezember 1998 Januar 2000 Author's address: Norbert Heuer Institut f¨urWissenschaftliche Datenverarbeitung Universit¨atBremen Postfach 33 04 40 D-28334 Bremen [email protected] http://www.iwd.uni-bremen.de/~heuer Contents 1 Introduction 1 2 Sobolev norms and polynomials 12 2.1 Discrete Sobolev inequalities . 20 2.2 Localization . 24 2.3 Discrete harmonic polynomials . 38 2.3.1 Representation of boundary element functions . 41 2.3.2 Extensions of boundary element functions . 42 3 Preconditioners 51 3.1 Additive Schwarz method . 54 3.2 Pseudo-differential operators of order one . 57 3.2.1 Multilevel additive Schwarz method for the h-version . 58 3.2.2 Overlapping additive Schwarz method for the p-version . 67 3.2.3 Iterative substructuring method for the p-version . 69 3.3 Pseudo-differential operators of order minus one . 75 3.4 Indefinite or non-selfadjoint pseudo-differential operators . 79 3.4.1 Additive Schwarz method . 80 3.4.2 Hybrid method . 89 3.4.3 Operators of order one . 93 3.4.4 Operators of order minus one . 98 3.5 Systems of pseudo-differential operators . 99 3.5.1 Indefinite systems . 102 3.5.2 Block skew-symmetric systems . 110 4 Examples 114 4.1 Helmholtz screen problems . 114 4.2 Electric screen problem . 132 4.3 Magnetic screen problem . 147 4.4 Helmholtz transmission problem . 152 4.5 Implementation issues . 162 Bibliography 169 iii List of Figures 2.1 Proof of Lemma 2.6: the domain R with Lipschitz continuous boundary. 19 2.2 Partition of unity: four typical cut-off functions. 26 1 2.3 The decomposition of Sh;p(Γ) induced by the localization operator Λ. 33 1 2.4 Decomposition of Sh;p(Γi0 ) for an L-point xi.................... 36 2.5 Discrete harmonic functions: an interior basis function for p = 8. 40 2.6 Discrete harmonic functions: an edge basis function for p = 8. 40 2.7 Discrete harmonic functions: vertex basis function for p = 8. 41 1 2.8 Extension of Sh;p(Γ): extended domain Ω with mesh Ωh. 43 3.1 Operators of order one: the mesh and its partition for the overlapping additive Schwarz method. 68 3.2 Operators of order one: the partition of the mesh for the wire basket precon- ditioner. 70 3.3 Operators of order one: the partition of the mesh for the non-overlapping additive Schwarz preconditioner. 71 3.4 Operators of order minus one: the partition of the mesh for the non-overlapping additive Schwarz preconditioner. 76 4.1 Example for the Dirichlet and Neumann problems for the Helmholtz operator: the screen Γ with uniform rectangular mesh. 119 1 4.2 Hypersingular operator (p-version, h− = 3): minimum eigenvalues obtained by the non-overlapping additive Schwarz preconditioner (H, NASM). 125 1 4.3 Weakly singular operator (p-version, h− = 2): minimum eigenvalues obtained by the non-overlapping additive Schwarz preconditioner (NASM). 127 1 4.4 Hypersingular operator (p-version, h− = 3): values of Λ0 and Λ1 obtained by the overlapping additive Schwarz preconditioner (S, OASM) with full local solvers (I) and positive definite local solvers (P). 129 1 4.5 Hypersingular operator (p-version, h− = 3): values of Λ0 and Λ1 obtained by the modified diagonal preconditioner (H, DIAG) with full local solvers (I) and positive definite local solvers (P). 131 1 4.6 Weakly singular operator (p-version, h− = 2): values of Λ0 and Λ1 obtained by the non-overlapping additive Schwarz preconditioner (NASM) with full local solvers (I) and positive definite local solvers (P). 132 1 4.7 Electric screen problem (k = 5, p-version, h− = 3): the ratios κ = Λ1=Λ0 obtained without and with different preconditioners. 141 iv LIST OF FIGURES v 1 4.8 Electric screen problem (p-version, h− = 3): values of Λ0 and Λ1 obtained by combining (NASM) and the overlapping additive Schwarz preconditioner (OASM,S) with full local solvers (I) and positive definite local solvers (P). 143 1 4.9 Electric screen problem (p-version, h− = 3): values of Λ0 and Λ1 obtained by combining (NASM) and the modified diagonal preconditioner (DIAG,H) with full local solvers (I) and positive definite local solvers (P). 144 1 4.10 Electric screen problem (k = 5, p-version, h− = 3): CPU-times for as- sembling the stiffness matrix (mat), the right hand side vector (rhs), for the iterative solution (sol) and for calculating the preconditioner (pre). The (NASM,I)/(H,DIAG,I)-preconditioner is used. 145 4.11 Electric screen problem (k = 5, h-version, p = 3): CPU-times for as- sembling the stiffness matrix (mat), the right hand side vector (rhs), for the iterative solution (sol) and for calculating the preconditioner (pre). The (NASM,I)/(H,DIAG,I)-preconditioner is used. 145 4.12 Electric screen problem (k = 5): relative error in energy norm versus total CPU-time (assembling stiffness matrix and right hand side, calculating the preconditioner (if present), and solving the linear system). 147 4.13 Example for the Helmholtz transmission problem in IR2: the L-shaped domain. 161 4.14 The grid used for the L-shaped domain. 162 4.15 Coupled FEM/BEM method for the Helmholtz transmission problem in IR2: the minimum eigenvalues of the symmetric parts of the system preconditioned by B. ........................................ 164 List of Tables 4.1 Theoretically justified preconditioners for the hypersingular operator Dk. 122 4.2 Hypersingular operator: condition numbers and extreme eigenvalues (standard basis functions (S), without and with overlapping additive Schwarz precondi- tioner (OASM)). 123 4.3 Hypersingular operator: condition numbers and extreme eigenvalues (discrete harmonic basis functions (H), with preconditioners: wire basket prec. (WIRE), non-overlapping additive Schwarz method (NASM), diagonal prec. (DIAG)). 124 4.4 Hypersingular operator: numbers of GMRES iterations (all cases). 125 4.5 Weakly singular operator: condition numbers and extreme eigenvalues (with- out and with non-overlapping additive Schwarz preconditioner (NASM)). 126 4.6 Weakly singular operator: numbers of GMRES iterations (without and with diagonal preconditioner (DIAG) and with non-overlapping additive Schwarz preconditioner (NASM)). 127 4.7 Hypersingular operator: numbers of GMRES iterations (without prec. (\{") and with overlapping additive Schwarz preconditioner (S, OASM), different wave numbers, full local solvers and positive definite local solvers (I, P)). 129 4.8 Hypersingular operator: numbers of GMRES iterations (with overlapping ad- ditive Schwarz preconditioner (S, OASM) and with modified diagonal precon- ditioner (H, DIAG), different wave numbers, full local solvers (I)). 131 4.9 Weakly singular operator: numbers of GMRES iterations (without prec. (\{") and with non-overlapping additive Schwarz preconditioner (NASM), different wave numbers, full local solvers and positive definite local solvers (I, P)). 133 4.10 Electric screen problem (k = 5): numbers of GMRES iterations (without prec. (\{") and with several preconditioners using full local solvers or positive definite local solvers (I, P)). 141 4.11 Electric screen problem (k = 10): numbers of GMRES iterations (without prec. (\{") and with several preconditioners using full local solvers or positive definite local solvers (I, P)). 142 4.12 Electric screen problem: numbers of GMRES iterations (without precondi- tioner and with (NASM)/(S,OASM) preconditioner (full local solvers) for dif- ferent wave numbers). 142 4.13 CPU-times for solving the electric screen problem (k = 5) up to an accuracy of 0:1 relative error in energy norm. 146 vi LIST OF TABLES vii 4.14 Coupled FEM/BEM method for the Helmholtz transmission problem in IR2: the minimum eigenvalues Λ0 of the symmetric parts, the norms Λ1 of the un-preconditioned and the preconditioned systems and the numbers #iter of iterations of the GMRES method. 163 Chapter 1 Introduction The most universal approach for approximately solving elliptic boundary value problems is by means of variational formulations over discrete function spaces, which is called Galerkin's method. There are several ways in deriving variational formulations. One can test the differential equation with suitable functions of appropriate Sobolev spaces by using the L2- inner product over the domain of the problem (more precisely the extension of the inner product by duality) and then by integrating by parts. Its solution within spaces of functions with small support is called finite element method (FEM), see, e.g., [6, 139, 32]. This method leads to large linear systems with sparse matrices and is most often used in practise. When considering problems on unbounded domains, e.g., scattering or transmission problem, this method is not directly applicable. In the case a fundamental solution of the problem exists, e.g., when the differential operator is linear with constant coefficients, an elegant way of solving is to represent its solution via Green's formula by an integral equation on the boundary of the problem. Then, testing with functions of appropriate Sobolev trace spaces leads to a variational formulation whose discretization needs only to consider the boundary of the domain. Here again, it is advantageous to take functions with small supports and then, the method is called boundary element method (BEM). However, since we now have to deal with integral operators which are non-local the arising linear systems are not sparse but in general fully occupied. On the other hand, since only the boundary needs to be discretized, the spatial dimensionality of the problem is reduced by one. Therefore, the linear systems of the BEM are not as large as for the FEM for comparable situations.