Boundary Element Methods: Foundation and Error Analysis

This chapter gives an expository introduction to the Galerkin-BEM for the elliptic boundary value problems from the mathematical point of view. Emphasis will be placed on the variational formulations of the boundary integral equations and the general error estimates for the approximate solution in appropriate Sobolev spaces. A classification of boundary integral equations will be given on the basis of the Sobolev index. The simple relations between the variational formulations of the boundary integral equations and the corresponding partial differential equations under consideration will be indicated. Basic concepts such as stability, consistency, and convergence, as well as the condition numbers and ill-posedness, will be discussed by using elementary examples. Keywords: boundary integral equations; fundamental solutions; variational formulations; Garding's inequality; Galerkin's method; boundary elements; stability; ill-posedness; asymptotic error estimates

[1]  J. Nitsche,et al.  Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens , 1968 .

[2]  N. Meyers,et al.  H = W. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[3]  J. Hess,et al.  Calculation of potential flow about arbitrary bodies , 1967 .

[4]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .

[5]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[6]  G. Hsiao.  On the stability of integral equations of the first kind with logarithmic kernels , 1986 .

[7]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[8]  Stefan Hildebrandt,et al.  Constructive proofs of representation theorems in separable Hilbert space , 1964 .

[9]  D. Hitchings,et al.  The nonlinear steady-state response of cable networks , 1976 .

[10]  W. Wendland.  Die Behandlung von Randwertaufgaben imR3 mit Hilfe von Einfach- und Doppelschichtpotentialen , 1968 .

[11]  F. Smithies,et al.  Singular Integral Equations , 1955, The Mathematical Gazette.

[12]  George C. Hsiao,et al.  A Galerkin collocation method for some integral equations of the first kind , 1980, Computing.

[13]  Zhonghai Ding,et al.  A proof of the trace theorem of Sobolev spaces on Lipschitz domains , 1996 .

[14]  F. D. Gakhov.  RIEMANN BOUNDARY VALUE PROBLEM , 1966 .

[15]  R. V. D. Geijn,et al.  A fast solution method for three‐dimensional many‐particle problems of linear elasticity , 1998 .

[16]  George C. Hsiao,et al.  Hybrid Coupled Finite-Boundary Element Methods for Elliptical Systems of Second Order , 2000 .

[17]  B. Silbermann,et al.  Numerical Analysis for Integral and Related Operator Equations , 1991 .

[18]  Kenichi Yoshida,et al.  Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D , 2001 .

[19]  W. Wendland.  Boundary Element Methods and Their Asymptotic Convergence , 1983 .

[20]  E. Sternberg,et al.  Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity , 1980 .

[21]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[22]  Olaf Steinbach,et al.  The construction of some efficient preconditioners in the boundary element method , 1998, Adv. Comput. Math..

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

[24]  S. Sirtori.  General stress analysis method by means of integral equations and boundary elements , 1979 .

[25]  Shmuel Agmon,et al.  Multiple layer potentials and the dirichlet problem for higher order elliptic equations in the plane I , 1957 .

[26]  G. Verchota.  Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains , 1984 .

[27]  Ernst P. Stephan,et al.  A hypersingular boundary integral method for two-dimensional screen and crack problems , 1990 .

[28]  J. Aubin.  Approximation of Elliptic Boundary-Value Problems , 1980 .

[29]  Olaf Steinbach,et al.  On the stability of the $L_2$ projection in fractional Sobolev spaces , 2001, Numerische Mathematik.

[30]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

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

[32]  Olaf Steinbach,et al.  Numerische Näherungsverfahren für elliptische Randwertprobleme , 2003 .

[33]  G. Hsiao.  On the Stability of Boundary Element Methods for Integral Equations of the First Kind , 1987 .

[34]  José M. Pérez-Jordá,et al.  A concise redefinition of the solid spherical harmonics and its use in fast multipole methods , 1996 .

[35]  Wolfgang L. Wendland,et al.  Some applications of a galerkin‐collocation method for boundary integral equations of the first kind , 1984 .

[36]  Richard C. MacCamy.  On a class of two-dimensional stokes flows , 1966 .

[37]  Wolfgang L. Wendland,et al.  Finite Element and Boundary Element Techniques from Mathematical and Engineering Point of View , 1988 .

[38]  Wolfgang L. Wendland,et al.  Singular perturbations for the exterior three-dimensional slow viscous flow problem , 1983 .

[39]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[40]  W. Wendland,et al.  A finite element method for some integral equations of the first kind , 1977 .

[41]  Rudolf Gorenflo,et al.  Conservative difference schemes for diffusion problems with boundary and interface conditions , 1980, Computing.

[42]  Martin Head-Gordon,et al.  Derivation and efficient implementation of the fast multipole method , 1994 .

[43]  Olaf Steinbach,et al.  Boundary element preconditioners for a hypersingular integral equation on an interval , 1999, Adv. Comput. Math..

[44]  Douglas N. Arnold,et al.  On the asymptotic convergence of collocation methods , 1983 .

[45]  Olaf Steinbach,et al.  Symmetric coupling of finite and boundary elements for exterior magnetic field problems , 2002 .

[46]  J. Nédélec.  Acoustic and electromagnetic equations , 2001 .

[47]  G. Of,et al.  Applications of a fast multipole Galerkin in boundary element method in linear elastostatics , 2005 .

[48]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[49]  G. Hsiao,et al.  A gårding's inequality for variational problems with constraints , 1994 .

[50]  Friedel Hartmann,et al.  Introduction to Boundary Elements , 1989 .

[51]  M. N. Le Roux.  Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension $2$ , 1977 .

[52]  R. Kress.  Linear Integral Equations , 1989 .

[53]  Houde Han,et al.  The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity , 1994 .

[54]  Ernst P. Stephan,et al.  On the Dirichlet problem in elasticity for a domain exterior to an arc , 1991 .

[55]  Olaf Steinbach,et al.  A fast multipole boundary element method for a modified hypersingular boundary integral equation , 2003 .

[56]  Ernst P. Stephan,et al.  Boundary integral equations for screen problems in IR3 , 1987 .

[57]  Olaf Steinbach,et al.  On C. Neumann's Method for Second-Order Elliptic Systems in Domains with Non-smooth Boundaries , 2001 .

[58]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[59]  Martin Costabel,et al.  Duality estimates for the numerical solution of integral equations , 1989 .

[60]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[61]  George C. Hsiao,et al.  Wavelet Collocation Methods for a First Kind Boundary Integral Equation in Acoustic Scattering , 2002, Adv. Comput. Math..

[62]  K. Atkinson,et al.  Theoretical Numerical Analysis: A Functional Analysis Framework , 2001 .

[63]  G. Hsiao,et al.  Solution of Boundary Value Problems by Integral Equations of the First Kind , 1973 .

[64]  Carlos Alberto Brebbia,et al.  Boundary Elements: An Introductory Course , 1989 .

[65]  M. A. Jaswon,et al.  Integral equation methods in potential theory and elastostatics , 1977 .

[66]  P. K. Banerjee.  The Boundary Element Methods in Engineering , 1994 .

[67]  Olaf Steinbach,et al.  Stability Estimates for Hybrid Coupled Domain Decomposition Methods , 2003 .

[68]  Dan Givoli,et al.  Natural Boundary Integral Method and Its Applications , 2002 .

[69]  Ernst P. Stephan,et al.  Remarks to Galerkin and least squares methods with finite elements for general elliptic problems , 1976 .

[70]  Jianxin Zhou,et al.  Boundary element methods , 1992, Computational mathematics and applications.