### 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) | 1-25 |

Number of pages | 25 |

Journal | Advances in Computational Mathematics |

DOIs | |

State | Accepted/In press - Jan 24 2018 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

**Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions.** / O’Neil, Michael.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

AU - O’Neil, Michael

PY - 2018/1/24

Y1 - 2018/1/24

N2 - 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.

AB - 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.

KW - Calder’on projectors

KW - Integral equation

KW - Laplace-Beltrami

KW - Potential theory

KW - Surface PDEs

UR - http://www.scopus.com/inward/record.url?scp=85040930647&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040930647&partnerID=8YFLogxK

U2 - 10.1007/s10444-018-9587-7

DO - 10.1007/s10444-018-9587-7

M3 - Article

AN - SCOPUS:85040930647

SP - 1

EP - 25

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

ER -