BEM with linear complexity for the classical boundary integral operators

Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panel-clustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Low-order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns.

[1]  J. Nédélec,et al.  Integral equations with non integrable kernels , 1982 .

[2]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[3]  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 .

[4]  Christian Lage,et al.  Transformation of hypersingular integrals and black-box cubature , 2001, Math. Comput..

[5]  Wolfgang Dahmen,et al.  Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method , 2003, Math. Comput..

[6]  Stefan A. Sauter,et al.  Variable Order Panel Clustering , 2000, Computing.

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

[8]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[9]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[10]  Frank Johnson.  A General Panel Method for the Analysis and Design of Arbitrary Configurations in Incompressible Flo , 1980 .

[11]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[12]  Jens Markus Melenk,et al.  Approximation of Integral Operators by Variable-Order Interpolation , 2005, Numerische Mathematik.

[13]  V. Hutson.  Integral Equations , 1967, Nature.

[14]  J. Dicapua.  Chebyshev Polynomials , 2019, Fibonacci and Lucas Numbers With Applications.

[15]  S. Sauter,et al.  May the singular integrals in BEM be replaced by zero , 2005 .

[16]  R. Bank,et al.  An algorithm for coarsening unstructured meshes , 1996 .

[17]  C. Schwab,et al.  Quadrature for $hp$-Galerkin BEM in ${\hbox{\sf l\kern-.13em R}}^3$ , 1997 .

[18]  Christoph Schwab,et al.  Fully Discrete Multiscale Galerkin BEM , 1997 .

[19]  P. Yla-Oijala,et al.  Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions , 2003 .

[20]  Wolfgang Hackbusch,et al.  On the efficient use of the Galerkin-method to solve Fredholm integral equations , 1993 .

[21]  Michael Griebel,et al.  A Particle-Partition of Unity Method-Part III: A Multilevel Solver , 2002, SIAM J. Sci. Comput..

[22]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[23]  W. Hackbusch,et al.  Composite finite elements for the approximation of PDEs on domains with complicated micro-structures , 1997 .

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

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

[26]  W. Hackbusch,et al.  Finite elements on degenerate meshes: inverse-type inequalities and applications , 2005 .

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

[28]  Stefan A. Sauter,et al.  Efficient automatic quadrature in 3-d Galerkin BEM , 1998 .

[29]  Johannes Elschner,et al.  The double-layer potential operator over polyhedral domains II: Spline Galerkin methods , 1992 .

[30]  C. Schwab,et al.  Quadrature for hp-Galerkin BEM in lR3 , 1997 .