An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

Cancès, Clément; Pop, Sorin; Vohralı́k, Martin

doi:10.1090/s0025-5718-2013-02723-8

Cited by 42 publications

(68 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For nonlinear elliptic equations, such constructions are unified for different numerical methods in [43]. In the context of two-phase flows, the constructions of u α,hτ , α ∈ {n, w}, can be found in [31] for cell-centered finite volume methods, in [19] for vertex-centered finite volume methods, and in [39,40,8] for the discontinuous Galerkin method.…”

Section: Concept Of Application To Different Numerical Methodsmentioning

confidence: 99%

“…It is to be noted that these estimators take the same form as those obtained in [19] (therein, no simplifications have been made).…”

Section: Scheme Definitionmentioning

confidence: 96%

“…Our estimates give a guaranteed and fully and easily computable upper bound on the selected error measure, the dual norm of the residual augmented by the distance of the approximate global and complementary pressures to proper function spaces. Recall that such error measure leads to the energy error for linear problems (cf., e.g., [42]), and it is shown in [19] that this is an upper bound on the error between the exact and approximate saturations, global pressures, and complementary pressures for conforming discretizations. Our estimates also allow to distinguish, estimate separately, and compare different error components.…”

Section: Introductionmentioning

confidence: 99%

“…Examples of the application of this abstract framework to different discrete formulations, spatial and temporal discretizations, linearizations, and linear solvers are given in Section 6, with some further examples in [31,19]. In order to unify the presentation, we have chosen once and for all as the primary unknowns the saturation and pressure of the wetting phase.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Vohralı́k

Wheeler

2013

Comput Geosci

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Concept Of Application To Different Numerical Methodsmentioning

confidence: 99%

“…It is to be noted that these estimators take the same form as those obtained in [19] (therein, no simplifications have been made).…”

Section: Scheme Definitionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Vohralı́k

Wheeler

2013

Comput Geosci

Self Cite

View full text Add to dashboard Cite

show abstract

“…This duality technique is rather standard; see [9] and the references therein. Its origins can be traced back at least to the elliptic projection of Wheeler [53].…”

Section: A2 Proof Of Theorem 52mentioning

confidence: 99%

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

2014

Self Cite

View full text Add to dashboard Cite

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 ‡

show abstract

Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

View full text Add to dashboard Cite

Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

show abstract

An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

Cited by 42 publications

References 56 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

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

Finite Volume Methods: Foundation and Analysis

Contact Info

Product

Resources

About