曲边区域上的多边形网格间断有限元离散及其多重网格算法

本文利用多边形网格上的间断有限元方法离散二阶椭圆方程,在曲边区域上,采用多条直短边逼近曲边的以直代曲的策略,实现了高阶元在能量范数下的最优收敛.本文还将这一方法用于带曲边界面问题的求解,同样得到高阶元的最优收敛.此外我们还设计并分析了这一方法的linebreakW-cycle和Variable V-cycle多重网格预条件方法,证明当光滑次数足够多时,多重网格预条件算法一致收敛.最后给出了数值算例,证实该算法的可行性并验证了理论分析的结果.

POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER

The main purpose of this paper is to study the discontinuous Galerkin discretization of second-order elliptic partial differential equations on curved regions. We use multiple short edges to approximate the curved boundary and achieve the optimal convergence order in H1 norm for high-order elements. This method is also applied to the interface problem with curved interfaces and obtains the optimal convergence of high-order elements. Furthermore, we prove that the W-cycle and V-cycle multigrid preconditioners converge uniformly provided that the number of smoothing is sufficiently large. Finally, numerical results are presented to verify the correctness of the theoretical results.