This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigri...
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This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
By combining two-grid method with domain decomposition method, a new local and parallel finite element algorithm based on the partition of unity is proposed for the incompressible flows. The interesting points in this...
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By combining two-grid method with domain decomposition method, a new local and parallel finite element algorithm based on the partition of unity is proposed for the incompressible flows. The interesting points in this algorithm lie in (1) a class of partition of unity is derived by a given triangulation, which guides the domain decomposition (2) the globally fine grid correction step is decomposed into a series of local linearized residual problems on some subdomains and (3) the global continuous finite element solution is obtained by assembling all local solutions together using the partition of unity functions. Some numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm.
By combining the techniques of the two-grid method and the partition of unity, we derive two local and parallel finite element algorithms for the Stokes problem. The most interesting features of these algorithms are (...
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By combining the techniques of the two-grid method and the partition of unity, we derive two local and parallel finite element algorithms for the Stokes problem. The most interesting features of these algorithms are (1) the partition of unity technique introduces a framework for domain decomposition;(2) only a series of local residual problems need to be solved on these subdomains in parallel, meanwhile requiring very little communication;(3) a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions. The optimal error estimates in L-2 and energy norms are proved under some assumptions. Also, several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms.
In this paper, several local and parallel finite element algorithms are proposed and analyzed for the 2D/3D stationary inductionless incompressible magnetohydrodynamic (MHD) equations. The core concept is to guarantee...
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In this paper, several local and parallel finite element algorithms are proposed and analyzed for the 2D/3D stationary inductionless incompressible magnetohydrodynamic (MHD) equations. The core concept is to guarantee the charge-conservation property by choosing mixed finite element spaces in ������0(div, omega) x ������20(omega) to approximate (������, ������), meantime combining the idea of domain decomposition method to realize parallel operation. The characteristic of the proposed algorithms is that the computational complexity is greatly reduced while ensuring the accuracy of the numerical simulation. With the local a prior estimate as the technical means of theoretical analysis, we give the error estimates of the algorithms. Finally, several numerical experiments are presented to verify the theoretical results.
A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fin...
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A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions.
In this study, based on a combination of the two-grid method and the partition of unity-based domain decomposition method, we propose a new local and parallel finite element algorithm for the elliptic boundary value p...
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In this study, based on a combination of the two-grid method and the partition of unity-based domain decomposition method, we propose a new local and parallel finite element algorithm for the elliptic boundary value problem. The proposed method has three key features: (1) it inherits the flexibility and controllability of domain decomposition based on the partition of unity;(2) global fine grid correction is replaced by solving a series of locally defined approximate residual problems with homogeneous Dirichlet boundary conditions on some finer grids;(3) a global continuous finite element solution is constructed by solving a coarse grid correction problem and by assembling all the local solutions together using the partition of unity subordinate. Under appropriate assumptions, the optimal error estimates in L-2 and the energy norms are proved by new analytical results. In addition, several numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm. (C) 2015 Elsevier Inc. All rights reserved.
In this paper, a new type of local and parallel algorithm is proposed to solve nonlinear eigenvalue problem based on multigrid discretization. Instead of solving the nonlinear eigenvalue problem directly in each level...
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In this paper, a new type of local and parallel algorithm is proposed to solve nonlinear eigenvalue problem based on multigrid discretization. Instead of solving the nonlinear eigenvalue problem directly in each level mesh, our method converts the nonlinear eigenvalue problem in the finest mesh to a linear boundary value problem on each level mesh and some nonlinear eigenvalue problems on the coarsest mesh. Further, the involved linear boundary value problems are solved using the local and parallel strategy. As no nonlinear eigenvalue problem is being solved directly on the fine spaces, which is time-consuming, this new type of local and parallel multigrid method evidently improves the efficiency of nonlinear eigenvalue problem solving. We provide a rigorous theoretical analysis for our algorithm and present details on numerical simulations to support our theory.
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