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1.
Summary.
This paper is concerned with a high order convergent
discretization for the semilinear reaction-diffusion problem:
,
for , subject to ,
where .
We assume that on
, which
guarantees uniqueness of a solution to
the problem. Asymptotic properties of
this solution are discussed. We consider a
polynomial-based three-point
difference scheme on a simple piecewise
equidistant mesh of Shishkin type.
Existence and local uniqueness of a solution
to the scheme are analysed. We
prove that the scheme is almost fourth order
accurate in the discrete maximum
norm, uniformly in the perturbation parameter
. We present numerical
results in support of this result.
Received February 25, 1994 相似文献
2.
We consider first the initial-boundary value problem for the parabolic equation
相似文献
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When the streamlinediffusion finite element method isapplied to convectiondiffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Qrot1 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamlinediffusionnorm convergence of order h3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements. 相似文献
6.
Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit
layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce
accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified
SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it
yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the
severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results.
Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000 相似文献
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