Numerical solution of Helmholtz's equation by the capacitance matrix method

Journal Article · Thu Jul 01 04:00:00 EDT 1976 · Math. Comput.; (United States)

DOI:https://doi.org/10.2307/2005318· OSTI ID:7090008

Proskurowksi, W; Widlund, O

In recent years the usefulness of fast Laplace solvers has been extended to problems on arbitrary regions in the plane by the development of capacitance matrix methods. The solution of the Dirichlet and Neumann problems for Helmholtz's equation is considered. It is shown that, by an appropriate choice of the fast solver, the capacitance matrix can be generated quite inexpensively. An analogy between capacitance matrix methods and classical potential theory for the solution of Laplace's equation is explored. This analogy suggests a modification of the method in the Dirichlet case. This new formulation leads to well-conditioned capacitance matrix equations which can be solved quite efficiently by the conjugate-gradient method. A highly accurate solution can, therefore, be obtained at an expense which grows no faster than that for a fast Laplace solver on a rectangle when the mesh size is decreased. 2 figures, 8 tables.

• OSTI ID: 7090008
• Journal Information: Math. Comput.; (United States), Journal Name: Math. Comput.; (United States) Vol. 30:135; ISSN MCMPA
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
DIFFERENTIAL EQUATIONS
DIRICHLET PROBLEM
EQUATIONS
MATRICES
NUMERICAL SOLUTION
POISSON EQUATION