Mixed Finite Element Methods on Nonmatching Multiblock Grids

Arbogast, Todd; Cowsar, Lawrence C.; Wheeler, Mary F.; Yotov, Ivan

doi:10.1137/s0036142996308447

Cited by 218 publications

(242 citation statements)

References 23 publications

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“…The points or lines where the numerical solution is superclose to the exact solution may be used to improve the accuracy of the overall simulation. Various superconvergence results for MFE methods have been established for rectangular meshes [21,19,27,10,11,12,3,1] and general quadrilateral meshes [2,13].…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Berndt¹,

Lipnikov²,

Shashkov³

et al. 2005

SIAM J. Numer. Anal.

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Abstract. Superconvergence of the velocity is established for mimetic finite difference approximations of second-order elliptic problems over h 2 -uniform quadrilateral meshes. The superconvergence result holds for a full tensor coefficient. The analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space. The theoretical results are confirmed by numerical experiments.Key words. mixed finite element, mimetic finite difference, tensor coefficient, superconvergence AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30 DOI. 10.1137/040606831 1. Introduction. We consider the numerical approximation of a linear secondorder elliptic problem. In porous medium applications, this equation models single phase Darcy flow and is usually written as a first-order system for the fluid pressure p and velocity u:

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Section: Introductionmentioning

confidence: 99%

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Berndt¹,

Lipnikov²,

Shashkov³

et al. 2005

SIAM J. Numer. Anal.

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“…This is particularly useful in applications where the Lagrange multiplier represents an important physical quantity. Similar systems arise when using mortar elements [7,2,3,6] as part of a domain decomposition approach [34] to solving elliptic or parabolic systems.…”

Section: Introductionmentioning

confidence: 95%

Adjoint-Based a Posteriori Error Estimation for Coupled Time-Dependent Systems

Asner¹,

Tavener

Kay

2012

SIAM J. Sci. Comput.

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Abstract. We consider time-dependent parabolic problems coupled across a common interface which we formulate using a Lagrange multiplier construction and solve by applying a monolithic solution technique. We derive an adjoint-based a posteriori error representation for a quantity of interest given by a linear functional of the solution. We establish the accuracy of our error representation formula through numerical experimentation and investigate the effect of error in the adjoint solution. Crucially, the error representation affords a distinction between temporal and spatial errors and can be used as a basis for a blockwise time-space refinement strategy. Numerical tests illustrate the efficacy of the refinement strategy by capturing the distinctive behavior of a localized traveling wave solution. The saddle point systems considered here are equivalent to those arising in the mortar finite element technique for parabolic problems. 1. Introduction. We consider an adjoint-based a posteriori error estimator for parabolic problems that are coupled across a given interface and construct an adaptive space-time finite element scheme. A posteriori error estimation is commonly used in parabolic problems for adaptive mesh refinement and the different approaches are described in a detailed review paper [1], as well as in [4,5,14,20,24,35,39,40,41]. In particular, [14,15,16] develop a technique based on solving an adjoint problem to estimate the error in a quantity of interest given by a functional of the solution as opposed to the error in a norm of the solution.Error estimation for coupled problems solved using operator decomposition techniques was considered in [17,18,9,8]. While operator decomposition is traditionally viewed as easy and inexpensive to implement [19,28], its convergence is rarely guaranteed. In this paper we take the alternative, monolithic approach to solving coupled parabolic problems, which do not suffer from this problem. We introduce a Lagrange multiplier (mortar element) space on the interface between the component domains, resulting in a large-scale complex system, which can nevertheless be solved efficiently with special treatment [33,22]. Moreover, the presence of the Lagrange multiplier variable allows us to derive error estimates for a quantity of interest that is supported along the interface. This is particularly useful in applications where the Lagrange

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“…We require that the following condition be satisfied [2], where in this paper k Á k r,S is the usual Sobolev norm of H r (S) (we may drop r if r = 0 and S if S = D).…”

Section: The Finite Element Approximationmentioning

confidence: 99%

“…Using a substructuring domain decomposition algorithm introduced in [24], the linear system resulting from the mortar mixed finite element method (4.3) on each realization can be reduced to a coarse scale interface problem in the mortar space M H [47,2,3]. The interface problem can be solved efficiently in parallel via a Krylov space iterative method, with each iteration requiring the solution of subdomain problems.…”

Section: A Multiscale Domain Decomposition Formulationmentioning

confidence: 99%

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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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Mixed Finite Element Methods on Nonmatching Multiblock Grids

Cited by 218 publications

References 23 publications

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Adjoint-Based a Posteriori Error Estimation for Coupled Time-Dependent Systems

A multiscale preconditioner for stochastic mortar mixed finite elements

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