Finite element method with nonuniform mesh sizes for unbounded domains

Journal Article · Wed Apr 01 04:00:00 EST 1981 · Math. Comput.; (United States)

DOI:https://doi.org/10.1090/S0025-5718-1981-0606503-5· OSTI ID:6350258

Goldstein, C I

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
• OSTI ID: 6350258
• Journal Information: Math. Comput.; (United States), Journal Name: Math. Comput.; (United States) Vol. 36:154; ISSN MCMPA
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
BOUNDARY-VALUE PROBLEMS
ERRORS
FINITE ELEMENT METHOD
NUMERICAL SOLUTION
OPTIMIZATION