$H^1$-Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation

$H^1$-Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation

Year:    2013

East Asian Journal on Applied Mathematics, Vol. 3 (2013), Iss. 2 : pp. 154–170

Abstract

We obtain the coefficient matrices of the finite element (FE), finite volume (FV) and finite difference (FD) methods based on $P_1$-conforming elements on a quasi-uniform mesh, in order to approximately solve a boundary value problem involving the elliptic Poisson equation. The three methods are shown to possess the same $H^1$-stability and convergence. Some numerical tests are made, to compare the numerical results from the three methods and to review our theoretical results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.030513.200513a

East Asian Journal on Applied Mathematics, Vol. 3 (2013), Iss. 2 : pp. 154–170

Published online:    2013-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

Keywords:    Stefan problems Godunov method solidification enthalpy cryosurgery. Finite element method finite difference method finite volume method Poisson equation stability and convergence.