Some Advice on Writing Multigrid Solvers

One key recommendation for writing MG codes is to make the code as modular as possible, with separate routines for each operation. Once relaxation, restriction, defect correction, and interpolation codes are developed, these can be used repeatedly for a wide range of problems. Also, preplanning and organization before starting to write the code can save a great deal of time and effort. 

A next stage might be to adapt the Poisson code to run on a parallel cluster using Message Passing Interface (MPI) coding. A first step here is to partition the fine-scale domain into subdomains the communication between the subdomains is limited to the boundary areas. If higher order Laplacians are used, however, the communication overhead increases. 

One issue encountered repeatedly by the author is the treatment of boimd-aries. On several occasions, the solver apparently worked, but then stalled at some finite residual. Almost invariably, that stalling was traced back to a sHght error on the boundaries in one of the routines (restriction, interpolation, relaxation, etc). So a word of warning is If the solver stalls before complete convergence, look first at the boundaries and how they have been handled in the code  

The author has found several basic sources to be instructive, especially in the early stages of learning about multigrid techniques. This is strictly a personal bias, but the book by Briggs, Henson, and MeCormick and the early review studies by Brandt have been particularly helpful and are recommended. They lay out both the theory and applications in clear language and are less mathematically oriented than some other multigrid sources. 

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