A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Hilhorst, Danielle; Vohralı́k, Martin

doi:10.1016/j.cma.2010.08.017

Cited by 14 publications

(12 citation statements)

References 29 publications

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“…For continuously differentiable rates the convergence of (adaptive) finite volume discretizations is studied in [22,32]; see also [10] for the convergence of a finite volume discretization of a copper-leaching model. In a similar framework, discontinuous Galerkin methods are discussed in [39] and upwind mixed finite element methods (MFEMs) are considered in [11,12]; combined finite volume-mixed hybrid finite elements are employed in [20,21]. Non-Lipschitz, but Hölder continuous rates are considered using conformal finite element method (FEM) schemes in [4,5].…”

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confidence: 99%

Convergence Analysis of Mixed Numerical Schemes for Reactive Flow in a Porous Medium

Kumar¹,

Pop²,

Radu³

2013

SIAM J. Numer. Anal.

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confidence: 99%

Convergence Analysis of Mixed Numerical Schemes for Reactive Flow in a Porous Medium

Kumar¹,

Pop²,

Radu³

2013

SIAM J. Numer. Anal.

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“…We then propose an adaptive algorithm which uses these criteria while simultaneously performing the usual local mesh refinement and equilibration of the spatial and temporal errors. This algorithm is inspired from [34,35,40,36,6] and from the work [18,23,14,19]. We conclude Section 4 by proving that, under these criteria, our estimators are also efficient while representing a lower bound for the dual norm of the residual.…”

Section: Introductionmentioning

confidence: 99%

“…(7.4) for an example. In the spirit of [41,36,48] and [18,23,14], we also propose the usual space-time adaptivity:…”

Section: Balancing and Stopping Criteriamentioning

confidence: 99%

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014

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We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡

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“…Inexact Newton methods have been studied in, e.g., [11,59,35,17,36,30,29]. Herein, we build upon the results of [49,42,37,47,43] which give guaranteed and robust a posteriori estimates with error components distinction.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

Vohralı́k

Wheeler

2013

Comput Geosci

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International audienceThis paper develops a general abstract framework for a posteriori estimates for immiscible incompressible two-phase flows in porous media. We measure the error by the dual norm of the residual and, for mathematical correctness, employ the concept of global and complementary pressures in the analysis. Our estimators allow to estimate separately the different error components, namely the spatial discretization error, the temporal discretization error, the linearization error, the iterative coupling error, and the algebraic solver error. We propose an adaptive algorithm wherein the different iterative procedures (iterative linearization, iterative coupling, iterative solution of linear systems) are stopped when the corresponding errors do not affect significantly the overall error, and wherein the spatial and temporal errors are equilibrated. Consequently, important computational savings can be achieved while guaranteeing a user-given precision. The developed framework covers fully implicit, implicit pressure-explicit saturation, or iterative coupling formulations; conforming spatial discretization schemes such as the vertex-centered finite volume method or the finite element method, and nonconforming spatial discretization schemes such as the cell-centered finite volume method, the mixed finite element method, or the discontinuous Galerkin method; linearizations such as the Newton or the fixed-point one; and general linear solvers. Numerical experiments for a model problem are presented to illustrate the theoretical results. Only by stopping timely the linear and nonlinear solvers, speed-ups by a factor between 10 and 20 in terms of the number of total linear solver iterations are achieved

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A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Cited by 14 publications

References 29 publications

Convergence Analysis of Mixed Numerical Schemes for Reactive Flow in a Porous Medium

Convergence Analysis of Mixed Numerical Schemes for Reactive Flow in a Porous Medium

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

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