Iterative two-grid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problem

In this paper, we design and analyze the iterative two-grid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that is just like the classical iterative two-grid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Secondly, we designed another iterative two-grid method, which replace the semilinear term by using the corresponding first-order Taylor expansion. Specifically, we need to construct a suitable initial value, which can be sorted out from an auxiliary variational problem, for the second iterative method. We also provide the error estimates for the second iterative algorithm and present numerical experiments to illustrate the theoretical result.