Domain Decomposition by the Mortar Element Method

Bernardi, Christine; Maday, Yvon; Patera, Anthony T.

doi:10.1007/978-94-011-1810-1_17

Cited by 549 publications

(876 citation statements)

References 5 publications

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“…The V h -ellipticity of the bilinear symmetrical form a(., .) follows from standard results (see for example [4]). Therefore, the first argument u h solution to the problem (3.4) is unique.…”

Section: Or 2 the First Studies And Convergence Results Correspondinmentioning

confidence: 97%

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

et al. 2001

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Abstract. In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

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Section: Or 2 the First Studies And Convergence Results Correspondinmentioning

confidence: 97%

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

et al. 2001

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“…We consider substructuring preconditioners and we report our numerical experience on solving problem (2). Our experiments confirm the theory depicting polylogarithmic bound for the condition number of the preconditioned matrix.…”

Section: Introductionsupporting

confidence: 72%

Substructuring Preconditioners for the Bidomain Extracellular Potential Problem

Pennacchio

Simoncini

Numerical Mathematics and Advanced Applications

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Summary. We study the efficient solution of the linear system arising from the discretization by the mortar method of mathematical models in electrocardiology. We focus on the bidomain extracellular potential problem and on the class of substructuring preconditioners. We verify that the condition number of the preconditioned matrix only grows polylogarithmically with the number of degrees of freedom as predicted by the theory and validated by numerical tests. Moreover, we discuss the role of the conductivity tensors in building the preconditioner.

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“…However, we note that the approximation and consistency contribution to the error (in the error approximation) may be analyzed and bounded through Strang's Second Lemma [8]. In particular, in [5,Eq. 18], the consistency error is written as a boundary integral of the normal derivative of the exact error multiplied by jump terms,…”

Section: Step 1: Non-conforming Error Approximationmentioning

confidence: 99%

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Eftang

Patera

2013

Numerical Meth Engineering

106

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We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are i) efficient empirical port approximation spaces -the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system, and ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditionerthe error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.

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Domain Decomposition by the Mortar Element Method

Cited by 549 publications

References 5 publications

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

Substructuring Preconditioners for the Bidomain Extracellular Potential Problem

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

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