Analysis of a goal-oriented adaptive two-grid finite-element algorithm for semilinear elliptic problems

In this paper, we propose, analyze, and numerically validate a goal-oriented adaptive two-grid finite-element method for second-order semilinear elliptic problems. In this method, the (k + 1)th and the kth adaptive meshes are considered as the fine and coarse meshes. The proposed algorithm requires a one-step Newton correction for the primal problem, and applies a special treatment to the reaction term for the dual problem, which in turn leads to linear discrete primal and dual problems having the same coefficient matrix. Therefore, this algorithm is more efficient than goal-oriented adaptive finite-element methods based on the classical Newton iteration. We prove contraction properties of the primal quasi-error and the combined primal-dual quasi-error, from the latter of which the convergence theory of the proposed method is established, up to higher order primal L-2-norm error terms implicitly requiring the initial mesh to be sufficiently fine. Some numerical examples are shown to illustrate the effectiveness and efficiency of this algorithm.