An outline of a multigrid procedure We now give a short description of the principles of a two-stage multigrid

7.7.1 An outline of a multigrid procedure We now give a short description of the principles of a two-stage multigrid

procedure: Step 1: Fine grid iterations. Perform iterations on the finest grid with mesh

spacing h to generate an intermediate solution y h to system A h . x = b with true solution vector x. The number of iterations is chosen sufficiently large that the short-wavelength oscillatory component of the error is effectively reduced, but no attempt is made to eliminate the long-wavelength error

component. The residual vector r h for the solution on this mesh satisfies r h =b−A h . y h (see equation (7.33)) and the error vector e h is given by

e h =x−y h (see equation (7.34)). We have also established that the error and residual are related as follows: A h .e h =r h (see equation (7.35)).

232 CHAPTER 7 SOLUTION OF DISCRETISED EQUATIONS

Step 2: Restriction. The solution is transferred from the fine mesh with spac- ing h onto a coarse mesh with spacing ch, where c > 1. Due to the larger mesh spacing of the coarse mesh the long-wavelength error (on the fine mesh) appears as a short-wavelength error on the new mesh and will reduce rapidly. The process of transferring can be simplified if we use a coarse mesh with twice the mesh spacing of the fine mesh. Instead of solving for the solution

vector y ch we work with the error equation A ch .e ch =r ch on the coarse mesh starting with an initial guess of e ch = 0. To perform the solution process we need values of the residual vector and the matrix of coefficients. Given the values of r h on the fine mesh we must use a suitable averaging procedure to find the residual vector r ch on the coarse mesh. The coefficients of matrix A ch can be recomputed from scratch on the coarser mesh or evaluated from the fine mesh coefficient matrix A h using some form of averaging or interpola- tion technique. The cost per iteration on the coarser mesh is small, so we can afford to perform an adequate number of iterations to get a converged

solution of the error vector e ch .

Step 3: Prolongation. After obtaining the converged solution of error vector

e ch for the coarse mesh we need to transfer it back to the fine mesh, but note that we have fewer data than points in the fine mesh. We use a convenient interpolation operator (e.g. linear interpolation) to generate values for the

prolonged error vector e′ h at intermediate points in the fine mesh. Step 4: Correction and final iterations. Once we have calculated the prolonged

error vector e′ h we may correct the intermediate fine grid solution: y improved = y h + e′ h . Because the long-wavelength error has been eliminated, this improved solution is closer to the true solution vector x. However, several approximations were made, so we perform a few more iterations with the improved solution to iron out any errors that may have been introduced during restriction and prolongation.

The above description is for the two-stage procedure (one fine mesh, one coarse mesh). In practice, however, the restriction is carried out into a number of increasingly coarse levels. Then prolongation procedures are also performed at each stage back to the starting mesh.