A STUDY OF INCORPORATING THE MULTIGRID METHOD INTO THE 3-DIMENSIONAL FINITE-ELEMENT DISCRETIZATION - A MODULAR SETTING AND APPLICATION

Increasing the efficiency of solving linear/linearized matrix equation s is a key point to save computer time in numerical simulation, especi ally for three-dimensional problems. The multigrid method has been det ermined to be efficient in solving boundary-value problems. However, t his method is mostly linked to the finite difference discretization, r ather than to the finite element discretization. This is because the g rid relationship between fine and coarse grids was not achieved effect ively for the latter case. Consequently, not only is the coding compli cated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we p resent an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite el ement discretization, such that we can take advantage of using the mul tigrid method. To achieve a consistent approximation at each grid, we use A(2h) = I-h(2h) A(h) I-2h(h) and b(2h) = I-h(2h) b(h), Starting fr om the composed matrix equation of the finest grid, to prepare the mat rix equations for coarse grids. Such a process is implemented on an el ement level to reduce the computation to its minimum. To demonstrate t he performance, this approach has been used to adapt two existing thre e-dimensional finite element subsurface flow and transport models, 3DF EMWATER and 3DLEWASTE, to their multigrid version, 3DMGWATER and 3DMGW ASTE, respectively. Two example problems, one for each model, are cons idered for illustration. The computational result shows that the multi grid method can help solve the example problems very efficiently with our presented modular setting. (C) 1998 John Wiley & Sons, Ltd.