A study of incorporating the multigrid method into the three-dimensional finite element discretization: a modular setting and application

Increasing the e¦ciency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be e¦cient in solving boundary-value problems. However, this method is mostly linked to the Þnite di¤erence discretization, rather than to the Þnite element discretization. This is because the grid relationship between Þne and coarse grids was not achieved e¤ectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the Þnite element discretization. Here we present an approach to systematically prepare necessary information to relate Þne and coarse grids regarding the three-dimensional Þnite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h"I2h h Ah Ih 2h and b2h"I2h h bh, starting from the composed matrix equation of the Þnest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional Þnite element subsurface ßow and transport models, 3DFEMWATER and 3DLEWASTE, to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very e¦ciently with our presented modular setting