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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We are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual s...
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ISBN:
(纸本)9783642026775;9783642026768
We are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. Coupling the inequality constraints by logarithmic barrier functions involving a barrier parameter and the PDE by Lagrange multipliers, the KKT conditions for the resulting saddle point problem represent a parameter dependent nonlinear system. The efficient numerical solution relies on multilevel path-following predictor-corrector techniques with an adaptive choice of the continuation parameter where the discretization is taken care of by finite elements with respect to nested hierarchies of simplicial triangulations of the computational domain. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuring transforming null space iterations. As an application in life sciences, we consider the optimal shape design of capillary barriers in microfluidic biochips.
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.
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.
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.
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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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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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.
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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