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

Multigrid solution for the cauchy problem associated with helmholtz type equation on non uniform grids

“…In multigrid method, the rate of convergence is independent of the mesh size. This method is more effective for solving large scale of sparse linear systems obtained from the discretization of elliptic PDEs [15,27,16,28,29]. The main principle of multigrid method is to smooth the error on coarse grid level using basic iterative methods such as Jacobi or Gauss-Seidel method, etc.…”

Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems

“…Because the coefficient matrix of the algebraic system is generally not diagonally dominant, a hybrid biconjugate gradient stabilized method (BiCGStab(2)) [29] was used in [11] instead of the conventional iterative methods such as the Gauss-Seidel or successive over relaxation (SOR) methods. Since multigrid method has been developed for many years and its combination with HOC difference schemes [9,15,[21][22][23][24][25][26][27][28] on uniform grids has been proved to be a high efficient and effective iterative technique, its combination with the HOC difference schemes on nonuniform grids is a natural procedure.…”

“…The difficulty of developing multigrid method on nonuniform grids is that those projection and interpolation operators developed on uniform grids [27,28,30] are not suitable for using on nonuniform grids. In other words, inexact projection and interpolation operators will cause a significant decrease in the efficiency of the standard multigrid methods and the convergence rate will deteriorate considerably.…”

“…Especially in recent years, various multigrid approaches and convergence acceleration techniques have been proposed to solve 2D and 3D Poisson equations and convection diffusion equations discretized by the fourth order compact difference schemes [9,15,[21][22][23][24][25][26][27][28]. Since in the earlier HOC schemes, the grid step lengths of different coordinate directions are equal, same to the case where nonuniform grid has been used with transformation, most multigrid methods are implemented on equal meshsize grids [9,[21][22][23][24][25][26].…”

“…Almost meanwhile, Zhang with his co-authors [15] proposed a fourth order compact finite difference scheme with different meshsizes in different coordinate directions for solving the 2D convection diffusion equation and employed a multilevel local mesh refinement strategy to deal with the local singularity problem. And lately, Ge [28] generalized the two specialized multigrid methods for solving 2D Poisson equation in [27] to 3D case on unequal-meshsize-discretization grids. To the best of our knowledge, there is still no any report about the implementation of multigrid method based on the transformation-free HOC scheme on nonuniform grids, and our study in this paper may fill this gap.…”