This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency ...
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ISBN:
(纸本)9783642026775;9783642026768
This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular H(div) system. Numerical experiments also justify the robustness and efficiency of this scheme.
There are several ways to solve in parallel the evolution problem P(∂t, ∂1, ⋯, ∂d) u = f. Explicit time discretization is naturally parallel. Implicit time discretization + spatial domain decomposition. For the heat e...
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In this paper, we extend the class of plane wave discontinuous Galerkin methods for the two-dimensional inhomogeneous Helmholtz equation presented in Gittelson, Hiptmair, and Perugia [2007]. More precisely, we conside...
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ISBN:
(纸本)9783642026775;9783642026768
In this paper, we extend the class of plane wave discontinuous Galerkin methods for the two-dimensional inhomogeneous Helmholtz equation presented in Gittelson, Hiptmair, and Perugia [2007]. More precisely, we consider the case of numerical fluxes defined in mixed form, namely, numerical fluxes explicitly defined in terms of both the primal and the flux variable, instead of the primal variable and its gradient. In our error analysis, we rely heavily on the approximation results and inverse estimates for plane waves proved in Gittelson, Hiptmair, and Perugia [2007] and develop a new mixed duality argument.
A recent theoretical result on optimized Schwarz algorithms, demonstrated at the algebraic level, enables the modification of an existing Schwarz procedure to its optimized counterpart. In this work, it is shown how t...
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ISBN:
(纸本)9783642026775;9783642026768
A recent theoretical result on optimized Schwarz algorithms, demonstrated at the algebraic level, enables the modification of an existing Schwarz procedure to its optimized counterpart. In this work, it is shown how to modify a bilinear finite-element method based Schwarz preconditioning strategy originally presented in [6] to its optimized version. The latter is employed to precondition the pseudo-Laplacian operator arising from the spectral element discretization of the magnetohydrodynamic equations in Elsasser form.
We discuss some overlapping domain decomposition algorithms for solving sparse nonlinear system of equations arising from the discretization of partial differential equations. All algorithms are derived using the thre...
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ISBN:
(纸本)9783642026775;9783642026768
We discuss some overlapping domain decomposition algorithms for solving sparse nonlinear system of equations arising from the discretization of partial differential equations. All algorithms are derived using the three basic algorithms: Newton for local or global nonlinear systems, Krylov for the linear Jacobian system inside Newton, and Schwarz for linear and/or nonlinear preconditioning. The two key issues with nonlinear solvers are robustness and parallel scalability. Both issues can be addressed if a good combination of Newton, Krylov and Schwarz is selected, and the right selection is often dependent on the particular type of nonlinearity and the computing platform.
We derive and analyze new boundary element (BE) based finite element discretizations of potential-type, Helmholtz and Maxwell equations on arbitrary polygonal and polyhedral meshes. The starting point of this discreti...
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ISBN:
(纸本)9783642026775;9783642026768
We derive and analyze new boundary element (BE) based finite element discretizations of potential-type, Helmholtz and Maxwell equations on arbitrary polygonal and polyhedral meshes. The starting point of this discretization technique is the symmetric BE Domain Decomposition Method (DDM), where the subdomains are the finite elements. This can be interpreted as a local Trefftz method that uses PDE-harmonic basis functions. This discretization technique leads to large-scale sparse linear systems of algebraic equations which can efficiently be solved by Algebraic Multigrid (AMG) methods or AMG preconditioned conjugate gradient methods in the case of the potential equation and by Krylov subspace iterative methods in general.
The main approaches to simulate fluid flows in complex moving geometries, use either moving-grid or immersed boundary techniques [5, 6, 7]. This former type of methods imply re-meshing, which are expensive computation...
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A particular class of mechanical systems concerns diffuse non smooth problems for which unilateral conditions may occur within the whole studied domain. For instance, when contact and friction occur as interactions be...
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In this note, we extend the mathematical framework in [7] of barrier methods for state constrained optimal control problems with PDEs to a more general setting. In [7] we modelled the state equation by Ly = u with L a...
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