Abstract
In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the method by using symmetric biorthogonal wavelet transforms to define the requisite operators. Numerical examples are presented to demonstrate the effectiveness of the modified wavelet multigrid method for diffusion problems with highly oscillatory coefficients, as well as for advection-diffusion equations in which the advection is moderately dominant.
Highlights
It is well known that the multigrid method is very useful in increasing the efficiency of iterative methods used to solve systems of algebraic equations approximating partial differential equations
In [5], for example, it is shown that a wavelet coarse grid operator defined by a Schur complement provides a good approximation to the homogenized coarse grid operator, and, as stated in [1,3], homogenization has been used to improve convergence of multigrid methods for diffusion problems with periodic coefficients (e.g., [6,7,8,9]) because the homogenized operator provides a very good approximation of the important properties of the original fine grid operator
The main idea for the remainder of the method follows in a similar manner as the wavelet multigrid method using orthogonal wavelet transforms. computed, and the resulting matrix, First, j L denoted by j
Summary
It is well known that the multigrid method is very useful in increasing the efficiency of iterative methods used to solve systems of algebraic equations approximating partial differential equations. In [1,3], the author extended the results of this approach to two dimensions and to multiple-level multigrid, dropping the assumption of a symmetric fine grid operator. This approach was considered for several reasons. For all numerical results in this paper, the V-cycle multigrid method is used with one iteration of the smoother for the coarsening and the correcting phases
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