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: -Du = f in W/sup C/, u = g on partialW, partialu/partialr+1/r u = o(1/r) as r = Vertical BarxVertical Bar..-->..infinity, where W/sup C/ is the complement in R/sup 3/ (three-dimensional Eucidean space) of a bounded set W with smooth boundary partialW, f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artifical bound G/sub R/ near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with W/sup C/ is denoted by W/sub R//sup C/ and the given problem is replaced by -Du/sub R/ = f in W/sup C//sub R/, u/sub R/ = g on partialW, partialu/sub R//partialr+1/r u/sub R/ = 0 on G/sub R/. This problem is then solved approximately by the finite element method, resulting in an approximate solution u /sub R//sup h/ for each h>0. In order to obtain a reasonably small error for u-u /sub R//sup h/ = (u-u/sub R/)+(u/sub R/-u /sub R//sup 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 Ch/sup -3/ with C independent of h and R.
- Research Organization:
- Applied Mathematics Department, Brookhaven National Laboratory, Upton, New York 11973
- DOE Contract Number:
- DE-AC02-76H00016
- OSTI ID:
- 6350258
- Journal Information:
- Math. Comput.; (United States), Vol. 36:154
- Country of Publication:
- United States
- Language:
- English
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