A \(\tau \)-preconditioner for a non-symmetric linear system arising from multi-dimensional Riemann-Liouville fractional diffusion equation. (English) Zbl 1506.65054
Summary: In this paper, we study a \(\tau \)-preconditioner for non-symmetric linear system arising from a steady-state multi-dimensional Riemann-Liouville (RL) fractional diffusion equation. The generalized minimal residual (GMRES) method is applied to solve the preconditioned linear system. Theoretically, we show that the GMRES solver for the preconditioned linear system has a convergence rate independent of discretization stepsizes. To the best of our knowledge, this is the first iterative solver with stepsize-independent convergence rate for the non-symmetric linear system. The proposed \(\tau \)-preconditioner is diagonalizable by the sine transform matrix, thanks to which the matrix-vector multiplication in each iteration step can be fast implemented by the fast sine transform (FST). Hence, the total operation cost of the proposed solver for the non-symmetric problem is linearithmic. Numerical results are reported to show the efficiency of the proposed preconditioner.
MSC:
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
Keywords:
preconditioning; convergence of GMRES; non-symmetric linear system; fractional diffusion equationReferences:
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