An ℎ𝑝-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type

Gudi, Thirupathi; Nataraj, Neela; Pani, Amiya

doi:10.1090/S0025-5718-07-02047-9

An $hp$-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type
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by Thirupathi Gudi, Neela Nataraj and Amiya K. Pani;

Math. Comp. 77 (2008), 731-756

DOI: https://doi.org/10.1090/S0025-5718-07-02047-9

Published electronically: November 21, 2007

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Abstract:

In this paper, an $hp$-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On $hp$-quasiuniform meshes, using the Brouwer 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 broken $H^1$ norm and $L^2$ norm which are optimal in $h$, suboptimal in $p$ are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

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