## A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation. (arXiv:1808.02937v1 [math.NA])

We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel, we construct a hierarchical matrix (H-matrix) to represent the stiffness matrix of the SEM and provide error estimates verified numerically. We can solve efficiently the H-matrix approximation problem using a hierarchical LU decomposition method, which reduces the computational cost to \$O(R^2 N_d \log^2N) +O(R^3 N_d \log N)\$, where \$R\$ it is the rank of submatrices of the H-matrix approximation, \$N_d\$ is the total number of degrees of freedom and \$N\$ is the number of elements. However, we lose the high accuracy of the SEM. Thus, we solve the corresponding preconditioned system by using the H-matrix approximation problem as a preconditioner, recovering the high order accuracy of the SEM. The condition number of the preconditioned system is independent of the polynomial查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel, we construct a hierarchical matrix (H-matrix) to represent the stiffness matrix of the SEM and provide error estimates verified numerically. We can solve efficiently the H-matrix approximation problem using a hierarchical LU decomposition method, which reduces the computational cost to \$O(R^2 N_d \log^2N) +O(R^3 N_d \log N)\$, where \$R\$ it is the rank of submatrices of the H-matrix approximation, \$N_d\$ is the total number of degrees of freedom and \$N\$ is the number of elements. However, we lose the high accuracy of the SEM. Thus, we solve the corresponding preconditioned system by using the H-matrix approximation problem as a preconditioner, recovering the high order accuracy of the SEM. The condition number of the preconditioned system is independent of the polynomial