A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation

• Published in: arXiv.org
• Main Authors: Li, Xianjuan, Mao, Zhiping, Song, Fangying, Wang, Hong, Karniadakis, George Em
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.08.2018
• Subjects: Algorithms; Approximation; Computational efficiency; Differential equations; Kernels; Mathematical analysis; Polynomials; Reduction; Spectral element method; Stiffness matrix
• ISSN: 2331-8422

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 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 \(O(R^2 N_d \log^2 N)+O(R^3 N_d \log N)+O(N_d^2)\). Moreover, by using a structured mesh (uniform or geometric mesh), we can further reduce the computational cost to \(O(R^2 N_d\log^2 N) +O(R^3 N_d \log N)+ O(P^2 N\log N)\) for the preconditioned system. We present several numerical tests to illustrate the proposed algorithm using \(h\) and \(p\) refinements.