Multigrid Method and Fourth-Order Compact Scheme for 2D Poisson Equation with Unequal Mesh-Size Discretization

A fourth-order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two-dimensional Poisson equation in a rectangular domain. Multigrid methods using a partial semicoarsening strategy and line Gauss–Seidel relaxation are designed to solve the resulting sparse linear systems. Numerical experiments are conducted to test the accuracy of the fourth-order compact difference scheme and to compare it with the standard second-order difference scheme. Convergence behavior of the partial semicoarsening and line Gauss–Seidel relaxation multigrid methods is examined experimentally.