Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of $O(h^{-2})$ algebraic complexity and of high parallel efficiency, where $h$ denotes the usual discretization parameter.

[1]  Carsten Carstensen.  Nonlinear interface problems in solid mechanics -- Finite element and boundary element coupling , 1993 .

[2]  Ernst P. Stephan,et al.  On the convergence of the multigrid method for a hypersingular integral equation of the first kind , 1990 .

[3]  Carsten Carstensen,et al.  Interface problem in holonomic elastoplasticity , 1993 .

[4]  Wolfgang Hackbusch,et al.  On the efficient realization of sparse matrix techniques for integral equations with focus on panel clustering, cubature and software design aspects , 1997 .

[5]  Ulrich Langer,et al.  Parallel algorithms for symmetric boundary element equations , 1996 .

[6]  Tuomo Rossi,et al.  Two Iterative Methods for Solving the Stokes Problem , 1993 .

[7]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[8]  Martin Costabel,et al.  Boundary Integral Operators on Lipschitz Domains: Elementary Results , 1988 .

[9]  Martin Costabel,et al.  Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation , 1985 .

[10]  Wolfgang Dahmen,et al.  Multiscale methods for the solution of the Helmholtz and Laplace equations , 1997 .

[11]  Wolfgang L. Wendland.  Boundary Element Topics , 1997 .

[12]  W. Queck.  The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type , 1989 .

[13]  Carsten Carstensen,et al.  Interface problems in viscoplasticity and plasticity , 1994 .

[14]  Olaf Steinbach.  Gebietszerlegungsmethoden mit Randintegralgleichungen und effiziente numerische Lösungsverfahren für gemischte Randwertprobleme , 1996 .

[15]  Olaf Steinbach,et al.  FEM BEM - a parallel solver for linear and nonlinear coupled FE/BE-equations , 1994 .

[16]  D. Faddeev,et al.  Computational methods of linear algebra , 1981 .

[17]  A. A. Samarskii,et al.  Numerical Methods for Grid Equations , 2018 .

[18]  J. Pasciak,et al.  The analysis of multigrid algorithms for pseudodifferential operators of order minus one , 1994 .

[19]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[20]  Stefan M. Holzer.  On the Engineering Analysis of 2D Problems by the Symmetric Galerkin Boundary Element Method and Coupled BEM/FEM , 1993 .

[21]  Arnd Meyer,et al.  Hierarchical Extension Operators and Local Multigrid Methods in Domain Decomposition Preconditioners , 1994 .

[22]  Olof B. Widlund,et al.  A Domain Decomposition Algorithm Using a Hierarchical Basis , 1990, SIAM J. Sci. Comput..

[23]  On Multigrid Methods of the First Kind for Symmetric Boundary Integral Equations of Nonnegative Order , 1989 .

[24]  G. Schmidt.  Boundary element discretization of Poincar\'e--Steklov operators , 1994 .

[25]  Ernst P. Stephan,et al.  Multigrid solvers and preconditioners for first kind integral equations , 1992 .

[26]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[27]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[28]  Uwe J. Langer,et al.  Parallel iterative solution of symmetric coupled FE/BE-equa-tions via domain decomposition , 1994 .

[29]  Gundolf Haase.  Hierarchical extension operators plus smoothing in domain decomposition preconditioners , 1997 .

[30]  D. Faddeev,et al.  Computational Methods of Linear Algebra , 1959 .

[31]  M. Kreienmeyer,et al.  Different Methodologies for Coupled BEM and FEM with Implementation on Parallel Computers , 1997 .

[32]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[33]  Gundolf Haase,et al.  Adaptive Domain Decomposition Methods for Finite and Boundary Element Equations , 1997 .

[34]  C.-C. Jay Kuo,et al.  A Domain Decomposition Preconditioner Based on a Change to a Multilevel Nodal Basis , 1991, SIAM J. Sci. Comput..

[35]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .