Hongling Hu

Dongdong He, Kejia Pan, Hongling Hu

In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically verified.

Kejia Pan, Dongdong He, Hongling Hu

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 $L^{\infty}$-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.

Kejia Pan, Dongdong He, Hongling Hu

In this paper, we develop a new extrapolation cascadic multigrid (ECMG$_{jcg}$) method, which makes it possible to solve 3D elliptic boundary value problems on rectangular domains of over 100 million unknowns on a desktop computer in minutes. First, by combining Richardson extrapolation and tri-quadratic Serendipity interpolation techniques, we introduce a new extrapolation formula to provide a good initial guess for the iterative solution on the next finer grid, which is a third order approximation to the finite element (FE) solution. And the resulting large sparse linear system from the FE discretization is then solved by the Jacobi-preconditioned Conjugate Gradient (JCG) method. Additionally, instead of performing a fixed number of iterations as cascadic multigrid (CMG) methods, a relative residual stopping criterion is used in iterative solvers, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple Richardson extrapolation is used to cheaply get a fourth order approximate solution on the entire fine grid. Test results are reported to show that ECMG$_{jcg}$ has much better efficiency compared to the classical MG methods. Since the initial guess for the iterative solution is a quite good approximation to the FE solution, numerical results show that only few number of iterations are required on the finest grid for ECMG$_{jcg}$ with an appropriate tolerance of the relative residual to achieve full second order accuracy, which is particularly important when solving large systems of equations and can greatly reduce the computational cost. It should be pointed out that when the tolerance becomes smaller, ECMG$_{jcg}$ still needs only few iterations to obtain fourth order extrapolated solution on each grid, except on the finest grid. Finally, we present the reason why our ECMG algorithms are so highly efficient for solving such problems.