Discontinuous Galerkin Methods for Quasi-Linear Elliptic Problems of Nonmonotone Type

Authors:
Thirupathi Gudi;Amiya K. Pani
Affiliations:
-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2007

Citing 0
Cited 6

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

Journal of Scientific Computing
On L2-error estimate for nonsymmetric interior penalty Galerkin approximation to linear elliptic problems with nonhomogeneous Dirichlet data

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Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

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Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

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A Posteriori Error Estimates of Discontinuous Galerkin Method for Nonmonotone Quasi-linear Elliptic Problems

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Global superconvergence and a posteriori error estimates of the finite element method for second-order quasilinear elliptic problems

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Abstract

In this paper, both symmetric and nonsymmetric interior penalty discontinuous $hp$-Galerkin methods are applied to a class of quasi-linear elliptic problems which are of nonmonotone type. Using Brouwer’s fixed point theorem, it is shown that the discrete problem has a solution, and then, using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in the broken $H^1$-norm, which are optimal in $h$ and suboptimal in $p$, are derived. Moreover, on a regular mesh an $hp$-error estimate for the $L^2$-norm is also established. Finally, numerical experiments illustrating the theoretical results are provided.