The numerical solution of partial differential equations for the simulation of physical phenomena on memory-coupled multiprocessor systems is discussed. The multigrid methods used are well suited for the considered systems, which are based on the distributed reconfigurable multiprocessor kit DIRMU. The implementation of a multilevel nodal diffusion method on special ring configurations built with DIRMU is outlined. The particular iteration scheme employed in the nodal expansion method appears similarly effective in parallel and serial environments. A general approach for mapping multigrid algorithms onto nearest neighbor mesh configurations, called EGPA, is presented and communication mechanisms are explained. Measured speedups for Poisson's equation and the more complicated steady-state Stokes equation are given. For large problems, the speedup is roughly proportional to the number of processors.