Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

Research output: Contribution to journalArticle

Abstract

The Laplace-Beltrami problem ΔΓψ = f has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green’s function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calderón projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles.

Original languageEnglish (US)
Pages (from-to)1-25
Number of pages25
JournalAdvances in Computational Mathematics
DOIs
StateAccepted/In press - Jan 24 2018

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Laplace
Integral equations
Three-dimension
Integral Equations
Layer Potentials
Fast multipole Method
Laplace-Beltrami Operator
Differential Geometry
Projector
Green's function
Learning systems
Triangle
Machine Learning
Physics
Topology
Numerical Examples
Geometry
Demonstrate

Keywords

  • Calder’on projectors
  • Integral equation
  • Laplace-Beltrami
  • Potential theory
  • Surface PDEs

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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