### Abstract

The Laplace-Beltrami problem Δ_{Γ}ψ = 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 language | English (US) |
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Pages (from-to) | 1385-1409 |

Number of pages | 25 |

Journal | Advances in Computational Mathematics |

Volume | 44 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics