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Mathematics of Computation

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A numerical scheme based on
mean value solutions for
the helmholtz equation
on triangular grids

Authors: M. G. Andrade and J. B. R. do Val
Journal: Math. Comp. 66 (1997), 477-493
MSC (1991): Primary 35A40, 65N06; Secondary 35J25, 65N15, 65N22
MathSciNet review: 1401937
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Abstract: A numerical treatment for the Dirichlet boundary value problem on regular triangular grids for homogeneous Helmholtz equations is presented, which also applies to the convection-diffusion problems. The main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method. Besides, this classical method can be seen as a truncated series approximation to the proposed method. The method is developed from the analytical solutions for the Dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle, yielding $O(h^{4})$ accuracy. Some numerical examples are discussed and the results are compared with other methods, with a consistent advantage to the solution obtained here.

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Additional Information

M. G. Andrade
Affiliation: Depto. de Ciencias de Computacao e Estatistica, Instituto de Ciencias Matematica de Sao Carlos, Universidade de Sao Paulo, C.P. 668 - Sao Carlos - SP, 13.560-970 - Brasil

J. B. R. do Val
Affiliation: Depto. de Telemática, Fac. de Eng. Elétrica, Universidade Estadual de Campinas - UNICAMP, C.P. 6101, 13081-970 - Campinas - SP, Brasil

Keywords: Numerical solutions for partial differential equations, elliptic differential equations, Helmholtz equations, non-standard difference approximation, convection-diffusion equations
Received by editor(s): July 31, 1995
Additional Notes: This work was partially supported by CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico, grant number 300573/95-2(NV) and 300721/86-2(NV)
Article copyright: © Copyright 1997 American Mathematical Society