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Exact finite element solutions to the Helmholtz equation
Author(s) -
Daly P.,
Helps J. D.
Publication year - 1973
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620060409
Subject(s) - mathematics , helmholtz equation , mathematical analysis , boundary value problem , eigenvalues and eigenvectors , eigenfunction , finite element method , dirichlet boundary condition , physics , thermodynamics , quantum mechanics
For one‐, two‐ and three‐dimensional co‐ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O ( h 2 n ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.