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

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发表日期：

2024年7月12日

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Xuehua Yang,
Zhimin Zhang

摘要

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 $w^n_h$ $w_{h}^{n}$ and $v^n_h$ $v_{h}^{n}$ of $w(\cdot , t_n)$ $w (\cdot, t_{n})$ and $\varDelta w(\cdot , t_n)$ $Δ w (\cdot, t_{n})$ are constructed. The stability of $w^n_h$ $w_{h}^{n}$ and $v^n_h$ $v_{h}^{n}$ are proved, and the a priori bounds of $\Vert w^n_h\Vert $ $‖ w_{h}^{n} ‖$ and $\Vert v^n_h\Vert $ $‖ v_{h}^{n} ‖$ are established, remaining $\alpha $ $α$ -robust as $\alpha \rightarrow 1^{-}$ $α \to 1^{-}$ . Then, the error $\Vert w(\cdot , t_n)- w^n_h\Vert $ $‖ w (\cdot, t_{n}) - w_{h}^{n} ‖$ and $\Vert \varDelta w(\cdot , t_n)-v^n_h\Vert $ $‖ Δ w (\cdot, t_{n}) - v_{h}^{n} ‖$ are estimated with $\alpha $ $α$ -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 $ $α$ . Finally some numerical results are provided to support our theoretical findings.