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

Abstract. In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree $r\ge1$ for a class of quasi-linear elliptic problems in $\Omega \subset \mathbb{R}^2$. 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 $\mathbb{R}^d$, $d=2,3$ and use it to establish the convergence of the two-grid method for problems in $\Omega \subset \mathbb{R}^3$.