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