Numerical Solution of Boundary-Value Problems for Partial Differential Equations Using Multigrid Methods in Nonuniform Domains

Abstract

SUMMARY In all the numerical methods the error E=] U- u ] which is the absolute value of the difference between the exact solution L of the differential equation and the approximate solution u of the difference equation, has two components ; one with high-frequency (or oscillatory) and another with low-frequency (or smooth). All the iterative (relaxation) methods such as Jacobi, weighted Jacobi, Gauss• Seidel and successive over relaxation (8OR) are very effective in eliminating the oscillatory component of the error, While leaving the smooth component relatively unchanged. This means that all these iterative methods possess what is called the " smoothing property". Now, we car ask how all these iterative methods can bc suitable for eliminating the smooth component af the error ?7 To answer this question, we shall use what is called the multigrid method. The idea of the multigrid mcthod is the transferring of the smooth component of thc error from thc present fine-grid to another coarser grid by using what is called the restriction transferring operator denoted by 1;

where I and 2h are the mesh sizes of the fine and coarse-grids. On the coarse-grid this smooth component of the error bccomes oscillatory component and any of the previous iterative methods can eliminate it very well. Again, and by using what is called the interpolatlon transferring operator denoted by l~h' this compam:nt which is called the correction tem of the error can be transferred back from the coarse-grid to the above fine-grid. The correction term is added to the previous approximate solution computed on this fine• grid to obtain a best possible solution on this grid. The transferring of the oscillatory component of the error from tle fine-grid to the coarse• grid and using the iterative method to improving it