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


Authors: Thirupathi Gudi, Neela Nataraj and Amiya K. Pani
Journal: Math. Comp. 77 (2008), 731-756
MSC (2000): Primary 65N12, 65N15, 65N30
DOI: https://doi.org/10.1090/S0025-5718-07-02047-9
Published electronically: November 21, 2007
MathSciNet review: 2373177
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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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Additional Information

Thirupathi Gudi
Affiliation: Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076
Email: trpathi@math.iitb.ac.in

Neela Nataraj
Affiliation: Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076
Email: neela@math.iitb.ac.in

Amiya K. Pani
Affiliation: Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076
Email: akp@math.iitb.ac.in

DOI: https://doi.org/10.1090/S0025-5718-07-02047-9
Keywords: $hp$-finite elements, local discontinuous Galerkin method, second order quasilinear elliptic problems, error estimates, order of convergence
Received by editor(s): April 14, 2006
Received by editor(s) in revised form: February 23, 2007
Published electronically: November 21, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society