A fast wavelet-multigrid method to solve elliptic partial differential equations

摘要

In this paper, we present a wavelet-based multigrid approach to solve elliptic boundary value problems encountered in mathematical physics. The system of equations arising from finite difference discretization is represented in wavelet-basis. These equations are solved using multiresolution properties of wavelets characterized by sparse matrices having condition number O(1) together with a multigrid strategy for accelerating convergence. The filter coefficients of D2k, k = 2, 3, 4 from Daubechies family of wavelets are used to demonstrate the effectiveness and efficiency of the method. The distinguishing feature of the method is; it works as both solver and preconditioner. As a consequence, it avoids instability, minimizes error and speeds up convergence. Compared to the classical multigrid method, this approach requires substantially shorter computation time; at the same time meeting accuracy requirements. It is found that just one cycle is enough for the convergence of wavelet-multigrid scheme whereas normally 7–8 cycles are required in classical multigrid schemes to meet the same accuracy. Numerical examples show that, the scheme offers a fast and robust technique for elliptic pde’s.