The Finite Element Method With Nonuniform Mesh Sizes for Unbounded Domains

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: -Am ■ / in Qc, u = g on 3ß, -~+ u = ol ] as r — |jc| -» oo, where ßc is the complement in R 3 (three-dimensional Euclidean space) of a bounded set il with smooth boundary 3Í2, / and g are smooth functions, and / has bounded support. This problem is approximately solved by introducing an artificial boundary TR near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with Qc is denoted by QR and the given problem is replaced by 0. In order to obtain a reasonably small error for u — «¿ « (u uR) + (uR — uj}), 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 Ch'3 with C independent of h and R.