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Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems


Authors: Peter A. Markowich and Miloš A. Zlámal
Journal: Math. Comp. 51 (1988), 431-449
MSC: Primary 65N30
MathSciNet review: 930223
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Abstract: This paper is concerned with the analysis of a class of "special purpose" piecewise linear finite element discretizations of selfadjoint second-order elliptic boundary value problems. The discretization differs from standard finite element methods by inverse-average-type approximations (along element sides) of the coefficient function $ a(x)$ in the operator $ - \operatorname{div}(a(x)\,{\operatorname{grad}}\,u)$. The derivation of the discretization is based on approximating the flux density $ J = a\,{\operatorname{grad}}{\mkern 1mu} u$ by constants on each element. In many cases the flux density is well behaved (moderately varying) even if $ a(x)$ and $ u(x)$ are fast varying.

Discretization methods of this type have been used successfully in semiconductor device simulation for many years; however, except in the one-dimensional case, the mathematical understanding of these methods was rather limited.

We analyze the stiffness matrix and prove that--under a rather mild restriction on the mesh--it is a diagonally dominant Stieltjes matrix. Most importantly, we derive an estimate which asserts that the piecewise linear interpolant of the solution u is approximated to order 1 by the finite element solution in the $ {H^1}$-norm. The estimate depends only on the mesh width and on derivatives of the flux density and of a possibly occurring inhomogeneity.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1988-0930223-7
Article copyright: © Copyright 1988 American Mathematical Society