2018/08/08 by Li, Xianjuan, Mao, Zhiping, Song, Fangying +2
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1808.02937
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(R2 Nd log2N) +O(R3 Nd log N), where R it is the rank of submatrices of the H-matrix approximation, Nd 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 degree P and grows with the number of elements, but at modest values of the rank R is below order 10 in our experiments, which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system; this reduction is higher in the two-sided fractional derivative compared to one-sided fractional derivative. The corresponding cost is O(R2 Nd log2 N)+O(R3 Nd log N)+O(Nd2). Moreover, by using a structured mesh (uniform or geometric mesh), we can further reduce the computational cost to O(R2 Ndlog2 N) +O(R3 Nd log N)+ O(P2 Nlog N) for the preconditioned system. We present several numerical tests to illustrate the proposed algorithm using h and p refinements.