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

• Published in: Applied mathematics and computation 2007-02, Vol.185 (1), p.667-680
• Main Authors: Bujurke, N.M., Salimath, C.S., Kudenatti, R.B., Shiralashetti, S.C.
• Format: Article
• Language: English
• Subjects: Condition number; Discrete wavelet transform; Exact sciences and technology; Filter coefficients; Fourier analysis; Mathematical analysis; Mathematics; Multigrid method; Multiresolution analysis; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2006.07.074

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 D 2 k , 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.