A practical finite element approximation of a semi-definite Neumann problem on a curved domain |
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Authors: | John W Barrett Charles M Elliott |
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Institution: | (1) Department of Mathematics, Imperial College, SW7 London;(2) Department of Mathematics, Purdue University, 47907 West Lafayette, IN, USA;(3) Present address: School of Mathematics & Physical Sciences, Univ. of Sussex, BN1 9QH Brighton, UK |
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Abstract: | Summary This paper considers the finite element approximation of the semi-definite Neumann problem: – ·(![sgr](/content/q3h62507817km2w1/xxlarge963.gif) u)=f in a curved domain ![OHgr](/content/q3h62507817km2w1/xxlarge937.gif) ![sub](/content/q3h62507817km2w1/xxlarge8834.gif)
n
(n=2 or 3),
on ![pgr](/content/q3h62507817km2w1/xxlarge960.gif) and
, a given constant, for dataf andg satisfying the compatibility condition
. Due to perturbation of domain errors (![OHgr](/content/q3h62507817km2w1/xxlarge937.gif) ![rarr](/content/q3h62507817km2w1/xxlarge8594.gif)
h
) the standard Galerkin approximation to the above problem may not have a solution. A remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds. Using this approach we show that for a finite element space defined overD
h
, a union of elements, with approximation powerh
k
in theL
2 norm and with dist ( ,
h
) Ch
k
, one obtains optimal rates of convergence in theH
1 andL
2 norms whether
h
is fitted (
h
D
h
) or unfitted (
h
D
h
) provided the numerical integration scheme has sufficient accuracy.Partially supported by the National Science Foundation, Grant #DMS-8501397, the Air Force Office of Scientific Research and the Office of Naval Research |
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Keywords: | AMS(MOS): 65 N 30 CR: G1 8 |
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