We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the W-cycle algorithm is uniformly conver...
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We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the W-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both W-cycle and V-cycle algorithms.
We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are ...
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We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means the convergence rates of the methods are independent of the mesh size and mesh level, Moreover, our theoretical analysis and convergence results do not require regularity assumptions of the model problems. Numerical results are given to support our theoretical findings. (C) 2015 Elsevier Inc. All rights reserved.
The present paper analyzes a multigrid algorithm for the Crouzeix–Raviart discretization of the Poisson and Stokes equations in two and three dimensions. The central point is the construction of easily computable $L^...
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The present paper analyzes a multigrid algorithm for the Crouzeix–Raviart discretization of the Poisson and Stokes equations in two and three dimensions. The central point is the construction of easily computable $L^2 $-projections based on suitable quadrature rules for the transfer from coarse to fine grids and vice versa.
multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated. Accuracy properties of finite difference approximation are discussed and validated. Two multigr...
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multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated. Accuracy properties of finite difference approximation are discussed and validated. Two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies: semicoarsening and standard coarsening. The resulting multigrid algorithms show robustness with respect to changes of the value of nu, the weight of the cost of the control, is sufficiently small. Fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on nu and on the discretization parameters. Results of numerical experiments are reported that demonstrate sharpness of Fourier analysis estimates. A multigrid algorithm that solves optimal control problems with box constraints on the control is considered. (C) 2003 Elsevier B.V. All rights reserved.
By combining computation from several scales of mesh fineness, multigrid and multilevel methods can improve speed and accuracy in a wide variety of science and engineering applications. This tutorial sketches the hist...
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By combining computation from several scales of mesh fineness, multigrid and multilevel methods can improve speed and accuracy in a wide variety of science and engineering applications. This tutorial sketches the history of the techniques, explains the basics, and gives painters to the literature and current research.
The constrained smoother for solving the saddle point system arising from the constrained minimization problem is a relaxation scheme such that the iteration remains in the constrained subspace. A multigrid method usi...
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The constrained smoother for solving the saddle point system arising from the constrained minimization problem is a relaxation scheme such that the iteration remains in the constrained subspace. A multigrid method using constrained smoothers for saddle point systems is analyzed in this paper. Uniform convergence of two-level and W-cycle multigrid methods, with sufficient many smoothing steps and full regularity assumptions, are obtained for some stable finite element discretization of Stokes equations. For Braess-Sarazin smoother, a convergence theory using only partial regularity assumption is also developed. (C) 2015 Elsevier Ltd. All rights reserved.
The goal of this work is to study multigrid methods in connection with the numerical solution of elliptic problems in the exterior of a bounded domain. The numerical method consists of approximating the original probl...
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The goal of this work is to study multigrid methods in connection with the numerical solution of elliptic problems in the exterior of a bounded domain. The numerical method consists of approximating the original problem by one on a truncated domain of diameter R and imposing a simple local approximate boundary condition on the outer boundary. The resulting problem is discretized using the finite element method. R must be made sufficiently large to reduce the truncation error (due to the approximate boundary condition) to the level of the discretization error. This results in a very large number of unknowns (increasing like O(R3) in three dimensions), when a quasi-uniform mesh is used. In previous work by the author [Math. Comp., 36 (1981), pp. 387-404], it was shown that optimal error estimates hold with the number of unknowns independent of R using a mesh grading procedure in which the size of the elements are systematically increased as their distance from the origin increases. In the present paper it is shown that the multigrid convergence rate is independent of R using a mesh grading of this kind (with the number of unknowns increasing like log R). It is also shown that the optimal error estimates in [Math Comp., 36 (1981), pp. 387-404] can be extended to datum with unbounded support. On the other hand, the number of multigrid iterations is bounded by O(R2) when a quasi-uniform mesh is used. For three-dimensional problems, the computational cost is bounded by O(R5) using a quasi-uniform mesh and by O(log R) using the graded mesh. The multigrid analysis is formulated and analyzed in a variational framework using weighted Sobolev spaces.
To solve the symmetric variational problem: $u \in H\, a(u,v) = L(v),\forall v \in H$, multigrid methods are constructed using a sequence of nested subspaces, $U_k ,k = 0$ to m, of H. The convergence is studied relati...
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To solve the symmetric variational problem: $u \in H\, a(u,v) = L(v),\forall v \in H$, multigrid methods are constructed using a sequence of nested subspaces, $U_k ,k = 0$ to m, of H. The convergence is studied relatively to the “energy-norm”, i.e. in the Hilbert space $(H,a( \cdot , \cdot ))$ and is based on the 2-level case $(m = 1)$. For the $(m + 1)$-level case, rather sharp bounds for the convergence factor can be constructed recursively, and their nondependence on the number of levels is established under certain hypothesis.
The choice of multigrid method depends strongly on the type of discretization used and the problem formulation employed. This article gives an overview of multigrid methods for the Stokes equations, focusing on the sa...
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The choice of multigrid method depends strongly on the type of discretization used and the problem formulation employed. This article gives an overview of multigrid methods for the Stokes equations, focusing on the saddle-point problem and on stable discretizations for staggered and vertex-centered grids.
In this paper, we present total variation diminishing (TVD) multigrid methods for computing the steady state solutions for systems of hyperbolic conservation laws. An efficient multigrid method should avoid spurious n...
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In this paper, we present total variation diminishing (TVD) multigrid methods for computing the steady state solutions for systems of hyperbolic conservation laws. An efficient multigrid method should avoid spurious numerical oscillations. This can be achieved by designing methods which preserve monotonicity and TVD properties through the use of upwind interpolation and restriction techniques. Such multigrid methods have been developed for scalar conservation laws in the literature. However, for hyperbolic systems, the upwinding directions are not apparent. We generalize the upwind techniques to systems by formulating the interpolation as solving a local Riemann problem. Upwind biased restriction is performed on the positive and negative components of the residual. This idea stems from the fact that the flux vector can be split into positive and negative components. For two-dimensional systems, we introduce a novel coarsening technique and extend the upwind interpolation and restriction techniques, which together capture the characteristics of the underlying system of hyperbolic equations. We provide a theoretical analysis to show that our two level method is TVD for one-dimensional linear systems. We also prove that both the additive and multiplicative multigrid schemes are consistent and convergent for one-dimensional linear systems. We demonstrate the effectiveness of our method by numerical examples including Euler equations.
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