Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

Vohralı́k, Martin

doi:10.1007/s00211-008-0168-4

Cited by 55 publications

(50 citation statements)

References 54 publications

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“…In other numerical methods, obtaining u α,hτ ∈ P 0 τ (H(div, Ω)) satisfying (3.1) is possible by means of local postprocessing. In the context of linear elliptic equations, we refer the reader to [44,70] for cell-centered finite volume methods, to [52,2,41] for discontinuous Galerkin methods, and to [28,57,14,71] for vertex-centered finite volume and finite element methods. For nonlinear elliptic equations, such constructions are unified for different numerical methods in [43].…”

Section: Concept Of Application To Different Numerical Methodsmentioning

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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Section: Concept Of Application To Different Numerical Methodsmentioning

confidence: 99%

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

Vohralı́k

Wheeler

2013

Comput Geosci

Self Cite

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“…Inhomogeneous Dirichlet boundary conditions are set according to the solution; the error stemming from their discrete approximation is neglected. This solution has been studied previously in [28,31,32] and provides an excellent test for a posteriori error estimation and adaptive mesh refinement due to the singularity at the point (0, 0).…”

Section: Mortar Couplingmentioning

confidence: 99%

Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

Pencheva¹,

Vohralı́k²,

Wheeler³

et al. 2013

SIAM J. Numer. Anal.

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Abstract. We consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h, whereas the interfaces are partitioned by a mesh of much coarser size H, and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested.

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“…Here, we restrict ourself to adaptive methods for finite volume discretization in the context of flow and transport in porous media. Closely related to our work, we refer for instance to [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Mesh Refinement for a Finite Volume Method for Flow and Transport of Radionuclides in Heterogeneous Porous Media

Amaziane

Bourgeois

Fatini

2013

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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-Adaptive Mesh Refinement for a Finite Volume Method for Flow and Transport of Radionuclides in Heterogeneous Porous Media -In this paper, we consider adaptive numerical simulation of miscible displacement problems in porous media, which are modeled by single phase flow equations. A vertex-centred finite volume method is employed to discretize the coupled system: the Darcy flow equation and the diffusion-convection concentration equation. The convection term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion-dispersion term is discretized by piecewise linear conforming finite elements. We introduce two kinds of indicators, both of them of residual type. The first one is related to time discretization and is local with respect to the time discretization: thus, at each time, it provides an appropriate information for the choice of the next time step. The second is related to space discretization and is local with respect to both the time and space variable and the idea is that at each time it is an efficient tool for mesh adaptivity. An error estimation procedure

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Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

Cited by 55 publications

References 54 publications

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

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

Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

Adaptive Mesh Refinement for a Finite Volume Method for Flow and Transport of Radionuclides in Heterogeneous Porous Media

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