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

Congreve, Scott; Houston, Paul; Wihler, Thomas P. (2013). Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs. Journal of scientific computing, 55(2), pp. 471-497. New York, N.Y.: Plenum Press 10.1007/s10915-012-9644-1

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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(TH,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(Th,p); thereby, only a linear system of equations is solved on the richer space V(Th,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(TH,P) and V(Th,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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