A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains

We present a high-order boundary integral equation solver for 3D elliptic boundary value problems on domains with smooth boundaries. We use Nystrom's method for discretization, and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals. The overall asymptotic complexity of our method is O(N3/2), where N is the number of discretization points on the boundary of the domain, and corresponds to linear complexity in the number of uniformly sampled evaluation points. A kernel-independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators. We describe a high-order accurate method for evaluating the solution at arbitrary points inside the domain, including points close to the domain boundary. We demonstrate how our solver, combined with a regular-grid spectral solver, can be applied to problems with distributed sources. We present numerical results for the Stokes, Navier, and Poisson problems.

[1]  D. Zorin,et al.  A fast solver for the Stokes equations with distributed forces in complex geometries , 2004 .

[2]  William Gropp,et al.  PETSc 2.0 users manual , 2000 .

[3]  Andrei Khodakovsky,et al.  Globally smooth parameterizations with low distortion , 2003, ACM Trans. Graph..

[4]  O. Bruno,et al.  A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications , 2001 .

[5]  T. Cruse,et al.  Some notes on singular integral techniques in boundary element analysis , 1993 .

[6]  D. E. Cormack,et al.  A new concept in near-singular integral evaluation: the continuation approach , 1989 .

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

[8]  Christoph Schwab,et al.  Numerical Integration of Singular and Hypersingular Integrals in Boundary Element Methods , 1992 .

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

[10]  W. Dahmen.  Multiscale and Wavelet Methods for Operator Equations , 2003 .

[11]  J. Thomas Beale,et al.  A Method for Computing Nearly Singular Integrals , 2000, SIAM J. Numer. Anal..

[12]  Luiz C. Wrobel,et al.  Boundary Integral Methods in Fluid Mechanics , 1995 .

[13]  Lexing Ying,et al.  A simple manifold-based construction of surfaces of arbitrary smoothness , 2004, ACM Trans. Graph..

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

[15]  Y. Mukaigawa,et al.  Large Deviations Estimates for Some Non-local Equations I. Fast Decaying Kernels and Explicit Bounds , 2022 .

[16]  Jonathan J L Higdon,et al.  Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes , 1995 .

[17]  Kendall E. Atkinson,et al.  The Numerical Solution of Laplace’s Equation in Three Dimensions , 1982 .

[18]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[19]  Robert H. Davis,et al.  An Efficient Algorithm for Hydrodynamical Interaction of Many Deformable Drops , 2000 .

[20]  Peter Rex Johnston,et al.  Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals , 1999 .

[21]  Wolfgang Dahmen,et al.  A multiscale method for the double layer potential equation on a polyhedron. , 2005 .

[22]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[23]  Sangtae Kim,et al.  Microhydrodynamics: Principles and Selected Applications , 1991 .

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

[25]  Jacob K. White,et al.  Multiscale Bases for the Sparse Representation of Boundary Integral Operators on Complex Geometry , 2002, SIAM J. Sci. Comput..

[26]  J. Phillips,et al.  12. Multipole and Precorrected-FFT Accelerated Iterative Methods for Solving Surface Integral Formulations of Three-dimensional Laplace Problems , 1995, Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering.

[27]  Christian Lage,et al.  Wavelet Galerkin algorithms for boundary integral equations , 1997 .

[28]  Zohar Yosibash,et al.  p-FEM2000?International Conference onp andhp Finite Element Methods: Mathematics and Engineering Practice , 2002 .

[29]  Kendall E. Atkinson.  QUADRATURE OF SINGULAR INTEGRANDS OVER SURFACES , 2004 .

[30]  Ian H. Sloan,et al.  Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ${\mathbb R}^3$ , 2002, Numerische Mathematik.

[31]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[32]  Thomas Y. Hou,et al.  Boundary integral methods for multicomponent fluids and multiphase materials , 2001 .

[33]  David E. Keyes,et al.  Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering , 1995 .

[34]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[35]  J. Higdon,et al.  A spectral boundary element approach to three-dimensional Stokes flow , 1995, Journal of Fluid Mechanics.

[36]  Ivan G. Graham,et al.  Hybrid Galerkin boundary elements: theory and implementation , 2000, Numerische Mathematik.

[37]  Ronald R. Coifman,et al.  Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations , 1993, SIAM J. Sci. Comput..

[38]  E. Wagner.  International Series of Numerical Mathematics , 1963 .

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

[40]  Kendall Atkinson,et al.  Numerical integration on the sphere , 1982, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[41]  Norbert Heuer,et al.  A p‐adaptive algorithm for the BEM with the hypersingular operator on the plane screen , 2002 .

[42]  C. Pozrikidis.  Boundary Integral and Singularity Methods for Linearized Viscous Flow: Index , 1992 .

[43]  L. Greengard,et al.  Integral Equation Methods for Stokes Flow and Isotropic Elasticity in the Plane , 1996 .

[44]  D. Zorin,et al.  A kernel-independent adaptive fast multipole algorithm in two and three dimensions , 2004 .

[45]  Ivan G. Graham,et al.  A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems , 1998 .

[46]  A StalzerMark,et al.  Numerical Solution of the Helmholtz Equation in 2D and 3D Using a High-Order Nyström Discretization , 1998 .

[47]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[48]  Johannes Tausch.  Sparse BEM for potential theory and Stokes flow using variable order wavelets , 2003 .

[49]  Norbert Heuer,et al.  Exponential convergence of the hp-version for the boundary element method on open surfaces , 1999, Numerische Mathematik.

[50]  Lexing Ying,et al.  A New Parallel Kernel-Independent Fast Multipole Method , 2003, ACM/IEEE SC 2003 Conference (SC'03).

[51]  Johannes Tausch,et al.  A variable order wavelet method for the sparse representation of layer potentials in the non-standard form , 2004, J. Num. Math..

[52]  Carsten Ebmeyer,et al.  Error Estimates for a Class of Degenerate Parabolic Equations , 1998 .

[53]  A. Mayo.  The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions , 1984 .

[54]  N. Nishimura.  Fast multipole accelerated boundary integral equation methods , 2002 .

[55]  John J. Ottusch,et al.  Numerical Solution of the Helmholtz Equation in 2D and 3D Using a High-Order Nyström Discretization , 1998 .