An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation

摘要

In this paper, we develop an EXCMG method to solve the three-dimensionalPoisson equation on rectangular domains by using the compact finite difference(FD) method with unequal meshsizes in different coordinate directions. Theresulting linear system from compact FD discretization is solved by theconjugate gradient (CG) method with a relative residual stopping criterion. Bycombining the Richardson extrapolation and tri-quartic Lagrange interpolationfor the numerical solutions from two-level of grids (current and previousgrids), we are able to produce an extremely accurate approximation of theactual numerical solution on the next finer grid, which can greatly reduce thenumber of relaxation sweeps needed. Additionally, a simple method based on themidpoint extrapolation formula is used for the fourth-order FD solutions ontwo-level of grids to achieve sixth-order accuracy on the entire fine gridcheaply and directly. The gradient of the numerical solution can also be easilyobtained through solving a series of tridiagonal linear systems resulting fromthe fourth-order compact FD discretizations. Numerical results show that ourEXCMG method is much more efficient than the classical V-cycle and W-cyclemultigrid methods. Moreover, only few CG iterations are required on the finestgrid to achieve full fourth-order accuracy in both the $L^2$-norm and$L^{\infty}$-norm for the solution and its gradient when the exact solutionbelongs to $C^6$. Finally, numerical result shows that our EXCMG method isstill effective when the exact solution has a lower regularity, which widensthe scope of applicability of our EXCMG method.