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

Lexing Ying, George Biros, Denis Zorin

Research output: Contribution to journalArticlepeer-review


We present a high-order boundary integral equation solver for 3D elliptic boundary value problems on domains with smooth boundaries. We use Nyström'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.

Original languageEnglish (US)
Pages (from-to)247-275
Number of pages29
JournalJournal of Computational Physics
Issue number1
StatePublished - Nov 20 2006


  • Boundary integral equations
  • Fast Fourier transform
  • Fast multipole method
  • Fast solvers
  • Laplace equation
  • Navier equation
  • Nearly singular integrals
  • Nyström discretization
  • Singular integrals
  • Stokes equation

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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