Capacitance matrix methods for the Helmholtz equation on general three dimensional regions

Dianne P. O'leary, Olof Widlund

Research output: Contribution to journalArticle

Abstract

Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are also explored in detail. Several conjugate gradient methods are discussed for the solution of the capacitance matrix equation. A fast Poisson solver is developed which is numerically very stable even for indefinite Helmholtz equations. Variants thereof allow substantial savings in primary storage for problems on very fine meshes. Numerical results show that accurate solutions can be obtained at a cost which is proportional to that of the fast Helmholtz solver in use.

Original languageEnglish (US)
Pages (from-to)849-879
Number of pages31
JournalMathematics of Computation
Volume33
Issue number147
DOIs
StatePublished - 1979

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Helmholtz equation
Matrix Method
Capacitance
Helmholtz Equation
Three-dimensional
Conjugate gradient method
Siméon Denis Poisson
Iterative methods
Computer programming
Potential Theory
Hermann Von Helmholtz
Conjugate Gradient Method
Matrix Equation
Programming
Directly proportional
Mesh
Analogue
Iteration
Costs
Numerical Results

Cite this

Capacitance matrix methods for the Helmholtz equation on general three dimensional regions. / O'leary, Dianne P.; Widlund, Olof.

In: Mathematics of Computation, Vol. 33, No. 147, 1979, p. 849-879.

Research output: Contribution to journalArticle

O'leary, Dianne P. ; Widlund, Olof. / Capacitance matrix methods for the Helmholtz equation on general three dimensional regions. In: Mathematics of Computation. 1979 ; Vol. 33, No. 147. pp. 849-879.
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