Iterative Methods for the Solution of Elliptic Problems on Regions Partitioned into Substructures

Bjørstad, Petter E.; Widlund, Olof B.

doi:10.1137/0723075

Cited by 367 publications

(238 citation statements)

References 22 publications

(16 reference statements)

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“…This approach is closely related (but more general) than the substructuring methods developed in e.g., [8,7,1]. If the basis functions for M H,i are chosen to be the Lagrange basis, then this procedure is similar to the construction of the local contribution to the Schur complement [39,36].…”

Section: End For End Formentioning

confidence: 99%

A multiscale preconditioner for stochastic mortar mixed finite elements

Wheeler

Wildey

Yotov

2011

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThe aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy's law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media.

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Section: End For End Formentioning

confidence: 99%

A multiscale preconditioner for stochastic mortar mixed finite elements

Wheeler

Wildey

Yotov

2011

Computer Methods in Applied Mechanics and Engineering

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“…For the numerical approximation of problem (34) the standard Lagrange finite elements can be used. It is well-known that for this kind of finite elements defined on a regular family of triangulations {T h } h>0 of Ω which induces a quasi-uniform family of triangulations on Γ, assumption H3 of Theorem 4.2 is satisfied with a constant independent of h (see, for instance, [8,11,19]). Moreover it is clear that assumptions H1 and H2 are satisfied with constants independent of h; hence, also for problem (34) we can prove the results reported in Theorems 5.1, 5.3 and 5.4.…”

Section: The Wave Problem In the Frequency Domainmentioning

confidence: 99%

Domain Decomposition Algorithms for Time-Harmonic Maxwell Equations with Damping

Rodríguez

Valli

2001

ESAIM: M2AN

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Abstract. Three non-overlapping domain decomposition methods are proposed for the numerical approximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). For each method convergence is proved and, for the discrete problem, the rate of convergence of the iterative algorithm is shown to be independent of the number of degrees of freedom.Mathematics Subject Classification. 65N55, 65N30.

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“…But in general, it is more effective to choose a l in each step according to a conjugate gradient algorithm (see, e.g., [6,7]). …”

Section: Error Propagationmentioning

confidence: 99%

Solving dynamic contact problems with local refinement in space and time

Hager

Hauret

Tallec

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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International audienceFrictional dynamic contact problems with complex geometries are a challenging task from the compu tational as well as from the analytical point of view since they generally involve space and time multi scale aspects. To be able to reduce the complexity of this kind of contact problem, we employ a non conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local struc ture and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme

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Iterative Methods for the Solution of Elliptic Problems on Regions Partitioned into Substructures

Cited by 367 publications

References 22 publications

A multiscale preconditioner for stochastic mortar mixed finite elements

A multiscale preconditioner for stochastic mortar mixed finite elements

Domain Decomposition Algorithms for Time-Harmonic Maxwell Equations with Damping

Solving dynamic contact problems with local refinement in space and time

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