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The finite element method with nonuniform mesh sizes for unbounded domains


Author: C. I. Goldstein
Journal: Math. Comp. 36 (1981), 387-404
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1981-0606503-5
MathSciNet review: 606503
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Abstract: The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem:

$\displaystyle - \Delta u = f\quad {\text{in}}\;{\Omega ^C},\quad u = g\quad {\t... ...= o\left( {\frac{1}{r}} \right)\quad {\text{as}}\;r = \vert x\vert \to \infty ,$

where $ {\Omega ^C}$ is the complement in $ {R^3}$ (three-dimensional Euclidean space) of a bounded set $ \Omega $ with smooth boundary $ \partial \Omega $, f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary $ {\Gamma _R}$ near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with $ {\Omega ^C}$ is denoted by $ \Omega _R^C$ and the given problem is replaced by

$\displaystyle - \Delta {u_R} = f\quad {\text{in}}\;\Omega _R^C,\quad {u_R} = g\... ...ial {u_R}}}{{\partial r}} + \frac{1}{r}{u_R} = 0\quad {\text{on}}\;{\Gamma _R}.$

This problem is then solved approximately by the finite element method, resulting in an approximate solution $ u_R^h$ for each $ h > 0$. In order to obtain a reasonably small error for $ u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h)$, it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by $ C{h^{ - 3}}$ with C independent of h and R.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0025-5718-1981-0606503-5
Article copyright: © Copyright 1981 American Mathematical Society

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