Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation

A fourth-order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three-dimensional (3D) Poisson equation on a cubic domain. Two multgrid methods are developed to solve the resulting sparse linear systems. One is to use the full-coarsening multigrid method with plane Gauss–Seidel relaxation, which uses line Gauss–Seidel relaxation to compute each planewise solution. The other is to construct a partial semi-coarsening multigrid method with the traditional point or plane Gauss–Seidel relaxations. Numerical experiments are conducted to test the computed accuracy of the fourth-order compact difference scheme and the computational efficiency of the multigrid methods with the fourth-order compact difference scheme.

Research highlights▶ The full-coarsening multigrid method employed to solve 2D Poisson equation in reference is generalized to 3D. ▶ A partial semi-coarsening multigrid method is developed to solve 3D Poisson equation. ▶ The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. ▶ The strategy can also be generalized to solve other 3D differential equations.