An Extrapolation Cascadic Multigrid Method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

Abstract

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the L 2 -norm and L ∞ -norm for the solution and its gradient when the exact solution belongs to C 6 . Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection–diffusion equations

“…Similar to FMG method, the EXFMG method starts with the coarsest grid to compute the solution by a direct solver, and applies an interpolation operator on the current grid to provide an initial guess for Algorithm 1 on the next finer grid. Besides, EXFMG method employs a fine mesh Richardson extrapolation technique [33] to get an extrapolation solution on each grid, when the two solutions with fourth-order accuracy on current and previous grids has been obtained by the Algorithm 1. The detailed algorithm is stated in Algorithm 3, where FMRE denotes a fine mesh Richardson extrapolation operator which will be defined and discussed in Sect.…”

“…They used a V-cycle multigrid method to get the fourth-order accurate solutions on both the fine and the coarse grids first, and then chose the iterative operator with Richardson extrapolation technique to compute the sixth-order accurate solution on the fine grid. We extend the idea described in the literature [32,33,39] to the original full multigrid method, and develop an extrapolation full multigrid (EXFMG) method to solve the 2D convection-diffusion equation with a fourth-order compact difference discretization. To be more precise, a bi-quartic Lagrange interpolation operator on a coarser grid is applied to get a good initial guess on the next finer grid for the multigrid V-or W-cycles solver.…”

“…In the classical Richardson extrapolation formula, two solutions on fine and coarse grids are used to obtain an extrapolation solution on the coarse grid. Based on the Richardson extrapolation technique and interpolation theory, in 2017, Pan proposed a mid-point extrapolation technique to enhance the accuracy of approximations on fine grid directly and cheaply (see [33] for details). Now, we will introduce this strategy for 2D case in the following.…”

“…Then the EXCMG method has been successfully applied to non-smooth elliptic problems [27,28], parabolic problems [29], and some other related problems [30][31][32]. Moreover, Pan [33] and Li [34,35] developed some EX-CMG methods combined with high-order compact difference schemes to solve Poisson equations. In 2017, Hu et al [36] obtained the super-optimality of the EXCMG method under the energy norm for H 2+α -regular (0 < α ≤ 1) elliptic problems.…”

Unique solvability of the CCD scheme for convection–diffusion equations with variable convection coefficients

“…Our future works will be focused on the convergence analysis of the CCD method and the generalization of the method for fractional order cases. And the CCD method for solving multi-dimensional elliptic boundary value problems [39][40][41][42] is also our future objective.…”