, Volume 49, Issue 3, pp 311-331
Date: 13 Jan 2011

Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

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In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r≥1 for a class of quasi-linear elliptic problems in Ω⊂ℝ2. We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H 1-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in Ω⊂ℝ3.

The research of C. Bi was partially supported by Shandong Province Natural Science Foundation (ZR2010AM004), a Project of Shandong Province Higher Educational Science and Technology Program (J09LA01, J10LA01), and the AMSS-PolyU Joint Research Institute for Engineering and Management Mathematics. The research of V. Ginting was partially supported by the National Science Foundation (DMS-1016283).