Multi-Grid Methods and Applications (Springer Series in Computational Mathematics)

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Maxim Larin and Arnold Reusken , A comparative study of efficient iterative solvers for generalized Stokes equations , Numer. Linear Algebra Appl. Olshanskii and A. Reusken , On the convergence of a multigrid method for linear reaction-diffusion problems , Computing 65 , no. Maxim A. Torgeir Rusten , Panayot S. Vassilevski , and Ragnar Winther , Interior penalty preconditioners for mixed finite element approximations of elliptic problems , Math.

Rob Stevenson , Nonconforming finite elements and the cascadic multi-grid method , Numer. Vanka , Block-implicit multigrid solution of Navier-Stokes equations in primitive variables , J. Wesseling and C. Oosterlee , Geometric multigrid with applications to computational fluid dynamics , J. Numerical analysis , Vol.

VII, Partial differential equations. Xu , The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids , Computing 56 , no. Zulehner , A class of smoothers for saddle point problems , Computing 65 , no. Arnold, F. Brezzi, and M. Fortin, A stable finite element method for the Stokes equations , Calcolo, 21 , pp. MR 86m 2. Bank, B. Welfert, and H. Yserentant, A class of iterative methods for solving saddle point problems , Numer.

Benzi, G.

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Golub, and J. Liesen, Numerical solution of saddle point problems , Acta Numerica, 14 , pp. MR m 4. Pironneau, Error estimates for finite element method solution of the Stokes problem in primitive variables Numer. MR 81g 5. Braess and W. Dahmen, A cascadic multigrid algorithm for the Stokes equations , Numer. MR c 6. MR 97k 7. Pasciak, Iterative techniques for time dependent Stokes problems, Comput.

MR 98e 8. Brenner, A nonconforming multigrid method for the stationary Stokes problem, Math. MR 91d 9. Bramble, J. Pasciak and O. Steinbach, On the stability of the projection in , Math. MR h Brezzi and M. MR 92d Handbook of Numerical Analysis, Vol. II, P. Ciarlet and J.

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The procedure goes on recursively until the finest grid. Recently, the Richardson extrapolation technique has been applied in multigrid methods. One group of researchers used Richardson extrapolation to construct a good initial guess for the iterative solver. This method uses a new extrapolation formula to construct a quite good initial guess for the iterative solution on the next finer grid, which greatly improves the convergence rate of the original CMG algorithm see [ 24 — 26 ] for details. Then the EXCMG method has been successfully applied to non-smooth elliptic problems [ 27 , 28 ], parabolic problems [ 29 ], and some other related problems [ 30 — 32 ].

In , Hu et al. Another group of researchers made efforts to design an iterative procedure with Richardson extrapolation strategy to enhance the solution accuracy on the finest grid. Sun, Zhang and Dai developed a sixth-order FD scheme for solving the 2D convection—diffusion equation [ 37 , 38 ]. In their approach, the ADI method is applied to compute the fourth-order accurate solution on the fine and coarse grids, respectively, then the Richardson extrapolation technique and an operator-based interpolation scheme are employed in each ADI iteration to calculate the sixth-order accurate solution on the fine grid.

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. 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. Besides, a stopping criterion related to relative residual is used to conveniently obtain the numerical solution with the desired accuracy.

Iterative Solvers: Multigrid Method (Part 1)

Moreover, a mid-point extrapolation strategy is used to obtain cheaply and directly a sixth-order accurate solution on the entire fine grid from two fourth-order solutions on two different grids current fine and previous coarse grids. Finally, through two examples chosen from the literature, the computational efficiency of the EXFMG method is discussed in detail. The rest of the paper is organized as follows.

And the extrapolation full multigrid method is described in Sect.

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Numerical results are given in Sect. Finally, concluding remarks are provided in Sect. All these functions are assumed to be sufficiently smooth and have the necessary continuous partial derivatives up to certain orders.

This equation is used to describe many processes in fields of fluid dynamics. The fourth-order compact difference scheme for Eq. By applying the fourth-order compact scheme 2 , we obtain the corresponding sparse linear system,. The main feature behind the MG method is to divide the errors into high frequency components and low frequency components and eliminate them in different grids. To be more specific, the high frequency component errors are eliminated on the fine grid, while the remaining low frequency errors are eliminated by a recursive procedure on the coarse grid.

A cycle of Algorithm 1, means that it starts from the finest grid to the coarsest grid and back to the finest grid again. When Algorithm 1 is applied to the finest grid system 2 in the solution of which we are interested, it can be repeated until the accuracy of approximation on the finest grid is considered to be high enough. In fact, Algorithm 1 can be incorporated with the classical FMG method naturally. It is well known that FMG delivers an approximation up to discretization accuracy if the multigrid cycle converges satisfactorily and if the order of FMG interpolation is larger than the discretization order of the PDE [ 8 ].

In this section, we shall establish an extrapolation full multigrid EXFMG method to solve the 2D convection—diffusion equation 1 with the fourth-order compact discretization 2. 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 key ingredients of this new method which are the main differences between the existing FMG methods include:.

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A bi-quartic Lagrange interpolation operator on coarse grid is taken as a full interpolation to provide a good initial guess on the next finer grid for multigrid iteration. The FMG method only computes the fourth-order accuracy solution of each discretization system. But the extrapolation method based on the midpoint formula see line 5 in Algorithm 3 and Sect. 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 , 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. Based on the above conditions, we can estimate the computational work of EXFMG with L levels of grids for d -dimensional problems as follows:.

We also present the numerical errors and computational costs time and WU in Figs. Now we discuss those results in the following paragraphs.