Finite element approximation of the $p$-Laplacian
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- by John W. Barrett and W. B. Liu PDF
- Math. Comp. 61 (1993), 523-537 Request permission
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 \[ - \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) = f\quad {\text {in}}\;\Omega \subset {\mathbb {R}^2},\quad u = g\quad {\text {on}}\;\partial \Omega .\] 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.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Math. Comp. 61 (1993), 523-537
- MSC: Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1993-1192966-4
- MathSciNet review: 1192966