Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

Abstract

A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk TWO-GRID hp-VERSION DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS FOR SECOND-ORDER QUASILINEAR ELLIPTIC PDESSCOTT CONGREVE, PAUL HOUSTON, AND THOMAS P. WIHLERAbstract. In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space V (T H , P ). The resulting 'coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space V (T h , p); thereby, only a linear system of equations is solved on the richer space V (T h , p). In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces V (T H , P ) and V (T h , p), respectively, in an automatic fashion. Numerical experiments are presented for both two-and three-dimensional problems; in each case, we demonstrate that the cpu time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method.

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