A new spectral boundary integral collocation method for three-dimensional potential problems

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30 Citations (Scopus)

Abstract

In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces Γ ⊂ R3. Under the assumption that Γ is diffeomorphic to the unit sphere ∂B, the original equation is transferred to an equivalent one on ∂B which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.

Original languageEnglish
Pages (from-to)778-805
Number of pages28
JournalSIAM Journal on Numerical Analysis
Volume35
Issue number2
DOIs
Publication statusPublished - 1998

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Potential Problems
Boundary Integral
Integral Method
Collocation Method
Collocation
Interpolation
Three-dimensional
Boundary integral equations
Interpolate
Mathematical operators
Longitude
Potential Theory
Boundary Integral Equations
Unit Sphere
Basis Functions
Rate of Convergence
Degree of freedom
Numerical Experiment
Equivalence
Projection

Keywords

  • Boundary integral equation
  • Collocation
  • Fourier approximation
  • Pseudospectral method
  • Three-dimensional potential problems

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

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abstract = "In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces Γ ⊂ R3. Under the assumption that Γ is diffeomorphic to the unit sphere ∂B, the original equation is transferred to an equivalent one on ∂B which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.",
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T1 - A new spectral boundary integral collocation method for three-dimensional potential problems

AU - Ganesh, M.

AU - Graham, I. G.

AU - Sivaloganathan, J.

PY - 1998

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N2 - In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces Γ ⊂ R3. Under the assumption that Γ is diffeomorphic to the unit sphere ∂B, the original equation is transferred to an equivalent one on ∂B which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.

AB - In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces Γ ⊂ R3. Under the assumption that Γ is diffeomorphic to the unit sphere ∂B, the original equation is transferred to an equivalent one on ∂B which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.

KW - Boundary integral equation

KW - Collocation

KW - Fourier approximation

KW - Pseudospectral method

KW - Three-dimensional potential problems

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