Corpus ID: 237571604

Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects

  title={Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects},
  author={David B. Stein and Alex H. Barnett},
Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (eg, in complex fluids) or when the solution is needed in the bulk, nearly-singular integrals must be evaluated at many targets. We show that precomputing a linear map from surface density to an effective source representation renders this task highly efficient, in the common case… Expand


Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion
We introduce a quadrature scheme—QBKIX —for the ubiquitous high-order accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracyExpand
On the evaluation of layer potentials close to their sources
The paper focuses on the solution of the Dirichlet problem for Laplace's equation in the plane with a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadratures, highPolynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Expand
Quadrature by expansion: A new method for the evaluation of layer potentials
This paper presents a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Expand
High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane
Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, inExpand
Evaluation of Layer Potentials Close to the Boundary for Laplace and Helmholtz Problems on Analytic Planar Domains
  • A. Barnett
  • Mathematics, Computer Science
  • SIAM J. Sci. Comput.
  • 2014
A simple and efficient scheme for accurate evaluation up to the boundary for single- and double-layer potentials for the Laplace and Helmholtz equations, using surrogate local expansions about centers placed near the boundary. Expand
Spectrally Accurate Quadratures for Evaluation of Layer Potentials Close to the Boundary for the 2D Stokes and Laplace Equations
This paper creates a “globally compensated” trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves and builds accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. Expand
A Fast Accurate Boundary Integral Method for Potentials on Closely Packed Cells
Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates whenExpand
A fast direct solver for quasi-periodic scattering problems
This work presents an integral equation based solver with O ( N ) complexity, which handles such ill-conditioning, using recent advances in “fast” direct linear algebra to invert hierarchically the isolated obstacle matrix. Expand
Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme
A fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex---possibly nonsmooth---geometries in two dimensions and it is shown that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners. Expand
A fast algorithm with error bounds for Quadrature by Expansion
A modified algorithm for QBX is improved with a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions, which empirically has cost-per-accuracy comparable to prior approaches. Expand