Superconvergence Analysis of a Robust Orthogonal Gauss Collocation Method for 2D Fourth-Order Subdiffusion Equations

Abstract

In this paper, we study the orthogonal Gauss collocation method (OGCM) with an arbitrary polynomial degree for the numerical solution of a two-dimensional (2D) fourth-order subdiffusion model. This numerical method involves solving a coupled system of partial differential equations by using OGCM in space together with the L1 scheme in time on a graded mesh. The approximations \(w^n_h\) and \(v^n_h\) of \(w(\cdot , t_n)\) and \(\varDelta w(\cdot , t_n)\) are constructed. The stability of \(w^n_h\) and \(v^n_h\) are proved, and the a priori bounds of \(\Vert w^n_h\Vert \) and \(\Vert v^n_h\Vert \) are established, remaining \(\alpha \)-robust as \(\alpha \rightarrow 1^{-}\). Then, the error \(\Vert w(\cdot , t_n)- w^n_h\Vert \) and \(\Vert \varDelta w(\cdot , t_n)-v^n_h\Vert \) are estimated with \(\alpha \)-robust at each time level. In addition, superconvergence results of the first-order and second-order derivative approximations are proved. These new error bounds are desirable and natural, as that they are optimal in both temporal and spatial mesh parameters for each fixed \(\alpha \). Finally some numerical results are provided to support our theoretical findings.