基于优化分层网格的多尺度有限元求解二维奇异摄动的计算格式与效率分析

Computational Scheme and Efficiency Analysis of Multiscale Finite Elements on Optimally Graded Meshes for Two-dimensional Singularly Perturbed Problems

针对奇异摄动问题的二维对流扩散方程,应用多尺度有限元法在优化的分层网格上探究高效计算方案。多尺度有限元法仅需在粗网格求解子问题,详细给出了多尺度之间的数据映射关系,将相应的微观信息代入宏观尺度,用于求解降低规模的矩阵方程以节约计算资源。基于摄动系数迭代,形成自适应分层网格,能够有效地逼近奇异摄动的边界层。通过数学分析与数值实验,对比计算消耗和运行时间,验证了多尺度有限元法随着分层网格的加密,可以获得稳定、高阶、高效的一致收敛结果,凸显新方法的计算效率与应用优势。

As for a two-dimensional convection-diffusion equation in the singular perturbation,a novel multiscale finite element method based on the optimally graded meshes is proposed.The multiscale finite element method just solves the sub-problems on coarse meshes,and the data mapping relationship for related scales is provided in details and the microscopic information is inherited to the macroscopic level.Then the matrix is reduced and its matrix equation is ready for solving efficiently.Based on the perturbed parameter,an adaptively graded mesh is constructed from its iterative formula,and the meshes are capable of approximating the boundary layers effectively.Through mathematical analyses and numerical experiments,to contrast the computational cost and execution time,the multiscale strategy on the graded mesh is validated to be the stable,high-order and short-time uniform convergence.Its computational efficiency and application advantage are prominent.