Theoretical and Practical Aspects of Multigrid Methods in Boundary Element Calculations

In this paper multigrid methods are advocated for the fast solution of the large nonsparse systems of equations that occur in boundary-element methods. Multigrid methods combine relaxation schemes and coarse-grid corrections. Ample attention is given to the decomposition of the system matrix in order to obtain a relaxation scheme that reduces the high-frequency components of the iteration error. It is shown that the decomposition should take the edges of the boundary into account, because they have a strong influence on the smoothing property of the relaxation scheme. The practical aspects of the multigrid method are concerned with the use of the method in boundary element calculations. The choice of the coarse-grid operators, the interactions between the grids and the implementation of the algorithm are discussed. The theoretical investigations show that the multigrid method converges more rapidly as the number of boundary elements increases. This is illustrated for two plane problems: (1) potential flow around an aerofoil and (2) interior fundamental problem of elasticity.