Solving Differential Equations Based On Wavelet And Multi-grid Method

The multi-grid method is a very useful technique for solving the linear and nonlinear equation systems obtained by discretizing the differential equations. It can speed the iteration by using the different steps of the grid to meet the "convergence layer " of the iteration. However, in the multi-grid algorithm, it is a very difficult issue to select the appropriate layer of coarse mesh and transfer operator between the fine and coarse mesh. Because of these, the wavelet-based multi-grid method is presented by combining the multi-resolution analysis of wavelet with the multi-grid method. The scaling spaces serve as the coarse grids in the multi-grid method, the wavelet low-pass (or high-pass) filter and its conjugate transpose operator serve as the interpolation and a constraint operator in multi-grid method.In this paper, we firstly give the computational structure of the wavelet-based multi-grid algorithm, and then compare it with the traditional multi-grid algorithm. The numerical results show that the novel method is convergent faster and more accurate than the traditional one. Secondly, we provide the positive proportional relationship between the truncation error of the mesh layer and the amount of correction. So that, we can select the area with local refinement according to the corrector then combine the multi-grid and the adaptive wavelet algorithm to improve the computational efficiency further.