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Finite element approximation of the $ p$-Laplacian

Authors: John W. Barrett and W. B. Liu
Journal: Math. Comp. 61 (1993), 523-537
MSC: Primary 65N30
MathSciNet review: 1192966
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Abstract: In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given $ p \in (1,\infty )$, f, and g, find u such that

$\displaystyle - \nabla \cdot (\vert\nabla u{\vert^{p - 2}}\nabla u) = f\quad {\... ...\;\Omega \subset {\mathbb{R}^2},\quad u = g\quad {\text{on}}\;\partial \Omega .$

The finite element approximation is defined over $ {\Omega ^h}$, a union of regular triangles, yielding a polygonal approximation to $ \Omega $. For sufficiently regular solutions u, achievable for a subclass of data f, g, and $ \Omega $, we prove optimal error bounds for this approximation in the norm $ {W^{1,q}}({\Omega ^h}),q = p$ for $ p < 2$ and $ q \in [1,2]$ for $ p > 2$, under the additional assumption that $ {\Omega ^h} \subseteq \Omega $. Numerical results demonstrating these bounds are also presented.

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Article copyright: © Copyright 1993 American Mathematical Society