Extrapolation cascadicmultigrid (EXCMG) method with the conjugate gradient smoother is shown to be an efficient solver for large sparse symmetric positive definite systems resulting from linear finite element discret...
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Extrapolation cascadicmultigrid (EXCMG) method with the conjugate gradient smoother is shown to be an efficient solver for large sparse symmetric positive definite systems resulting from linear finite element discretization of second-order elliptic boundary value problems [Pan et al. J. Comput. Phys. 344 (2017) 499-515]. In this paper, we generalize the EXCMG method to solve a class of spatial fractional diffusion equations (SFDEs) with variable coefficients. Both steady-state and time-dependent problems are considered. First of all, space-fractional derivatives defined in Riemann-Liouville sense are discretized by using the weighted average of shifted Grunwald formula, which results in a fully dense and nonsymmetric linear system for the steady-state problem, or a semi-discretized ODE system for the time-dependent problem. Then, to solve the former problem, we propose the EXCMG method with the biconjugate gradient stabilized smoother to deal with the dense and nonsymmetric linear system. Next, such technique is extended to solve the latter problem since it becomes fully discrete when the Crank-Nicolson scheme is introduced to handle the temporal derivative. Finally, several numerical examples are reported to show that the EXCMG method is an efficient solver for both steady-state and time-dependent SFDEs, and performs much better than the V-cycle multigridmethod with banded-splitting smoother for time-dependent SFDEs [Lin et al. J. Comput. Phys. 336 (2017) 69-86]. (C) 2021 Elsevier Inc. All rights reserved.
In this paper, we investigate the effectiveness of the cascadic multigrid method applied to improved Picard iteration method for nonlinear problems arising in deforming variably saturated porous media. The finite elem...
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In this paper, we investigate the effectiveness of the cascadic multigrid method applied to improved Picard iteration method for nonlinear problems arising in deforming variably saturated porous media. The finite element method with 6-node triangular elements is applied to discretize the space and obtain nonlinear algebraic equations. Then the nonlinear iterative method is used for iterative solution. Since the classical nonlinear Picard iteration method (PI) can be slow to converge, two improved Picard iteration methods are proposed. One is the improved Picard method with iterations k on the coarse grid and number of multiple grids m (PI-MG(m, k)), and the other is the improved Picard method based on the cascadic multigrid method without parameter k (PI-NMG (m)). Three numerical examples are given to verify the effectiveness of the improved method. Results indicate that the convergence rate of PI is the slowest (10-25 nonlinear iterations per time step), followed by PI-MG(m, k) (3-5 nonlinear iterations), and PI-NMG(m) is the fastest (stable 2 nonlinear iterations). The computational ef-ficiency of PI-NMG(m) is improved by about 90% relative to PI and by about 70% relative to AR-PI. This method can be applied to large-scale numerical calculation of seepage and deformation in unsaturated deforming porous media.
In this paper, a type of accurate a posteriori error estimator is proposed for semilinear Neumann problem, which provides an asymptotic exact estimate for the finite element approximate solution. As its applications, ...
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In this paper, a type of accurate a posteriori error estimator is proposed for semilinear Neumann problem, which provides an asymptotic exact estimate for the finite element approximate solution. As its applications, we design two types of cascadic adaptive finite element methods for semilinear Neumann problem based on the proposed a posteriori error estimator. The first scheme is based on the Newton iteration, which needs to solve a linearized boundary value problem by some smoothing steps on each adaptive space. The second scheme is based on the multilevel correction method, which contains some smoothing steps for a linearized boundary value problem on each adaptive space and a solving step for semilinear Neumann equation on a low dimensional space. In addition, the proposed a posteriori error estimator provides the strategy to refine mesh and control the number of smoothing steps for both of the cascadic adaptive methods. Some numerical examples are presented to validate the efficiency of the proposed algorithms in this paper. (C) 2019 Elsevier Inc. All rights reserved.
In this paper, we study a numerical algorithm to find all solutions of Gelfand equation. By utilizing finite difference discretization, the model problem defined on bounded do-main with Dirichlet condition is converte...
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In this paper, we study a numerical algorithm to find all solutions of Gelfand equation. By utilizing finite difference discretization, the model problem defined on bounded do-main with Dirichlet condition is converted to a nonlinear algebraic system, which is solved by cascadic multigrid method combining with Newton iteration method. The key of our numerical method contains two parts: a good initial guess which is constructed via col-location technique, and the Newton iteration step is implemented in cascadic multigrid method. Numerical simulations for both one-dimensional and two-dimensional Gelfand equations are carried out which demonstrate the effectiveness of the proposed algorithm. We find that by using the symmetry property of equation, numerical solutions can be obtained by mirror reflection after solving model problem in a sub-domain. This will save considerable time consumption and storage cost in computational process of cascadic multigrid method.(c) 2022 Elsevier Inc. All rights reserved.
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