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

Abstract

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 $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ . The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ ; thereby, only a linear system of equations is solved on the richer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ . 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 $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ and $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ , 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.