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

In this paper, we develop a new extrapolation cascadic multigrid (ECMG) method to solve the 3D Poisson equation using the compact finite difference (FD) method. First, a 19-point fourth-order compact difference scheme with unequal meshsizes in different coordinate directions is employed to discretize the 3D Poisson equation on rectangular domains. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation, we are able to obtain a quite good initial guess of the iterative solution on the next finer grid, which is a fifth-order approximation to the fourth-order FD solution. The resulting large linear system is then solved by the Conjugate Gradient (CG) method using a relative residual stopping criterion in order to conveniently obtain the numerical solution with the desired accuracy. In addition, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two levels of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly, which is much simpler and more efficient than the existing methods. Besides the numerical solution of the Poisson equation, the gradient of the numerical solution is also easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretization of the derivatives. Numerical results show that the above ECMG method is much more efficient than the classical multigrid method. Moreover, only few CG iterations are required on the finest grid for the new method with an appropriate tolerance to achieve full fourth-order accuracy in both the L2 and L∞ norms for the solution and its gradient when the exact solution is smooth enough. Finally, numerical result from one non-smooth problem shows that the new ECMG method is still effective when the exact solution belongs to H5, and the result is consistent with the existing theoretical results.