Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem

Kumar, Sarvesh; Ruiz–Baier, Ricardo

doi:10.1007/s10915-015-9993-7

Cited by 18 publications

(14 citation statements)

References 46 publications

(65 reference statements)

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“…Mass-lumping flux-correction strategies targeted for CWC enforcement could be also incorporated without much effort (see e.g. [31]). Moreover, monotonicity properties (essential in avoiding spurious oscillations and non-physical concentrations) are not discussed within our theoretical analysis, but our computational experiments along with coercivity and discrete inf-sup conditions satisfied by the formulation may indicate that this property holds.…”

Section: Discussionmentioning

confidence: 99%

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

Bürger¹,

Kumar

Ruiz–Baier

2015

Journal of Computational Physics

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Section: Discussionmentioning

confidence: 99%

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

Bürger¹,

Kumar

Ruiz–Baier

2015

Journal of Computational Physics

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“…This operator connects the trial and test spaces, and it is self-adjoint with respect to the L 2 −inner product. Moreover the following result can be established (see [9,25,26] for a proof ).…”

Section: Meshes and Preliminariesmentioning

confidence: 93%

“…• Using an extension of the operator defined in (4.1), it is possible to construct a full DFVE scheme for the discretization of (2.3)-(2.6). This simply requires replacing (4.2) by a DFVE formulation of velocity and pressure as proposed for instance in [14,26] for Stokes-related problems. This can be done using either stabilized discontinuous Galerkin P 1 − P 1 approximations of velocity and pressure, or local pressure projection stabilizations.…”

Section: Statement Of the Combined Mixed-primal Schemementioning

confidence: 99%

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

Ruiz–Baier

Lunati

2016

Journal of Computational Physics

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We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for the mixture and for one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi-Douglas-Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the method can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two-and three-dimensional porous media are presented to demonstrate the accuracy of the method that displays an optimal convergence rate, the physics-preserving properties of the mixed-primal scheme, as well as the the robustness of the method, which is successfully applied to diverse physical phenomena such as density fingering, Terzaghi's consolidation, deformation of a cantilever bracket, and the Boycott effects. The applicability of the method is not limited to flow in porous media, but can be also employed to describe many other physical systems that are governed by a similar system of equations, e.g., to model multi-component materials.

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“…The application of these schemes in the approximation of Stokes and related fluid problems can be found in e.g. [ 10 – 12 , 24 , 32 , 33 , 54 ].…”

Section: Introductionmentioning

confidence: 99%

Error Bounds for Discontinuous Finite Volume Discretisations of Brinkman Optimal Control Problems

2018

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We introduce a discontinuous finite volume method for the approximation of distributed optimal control problems governed by the Brinkman equations, where a force field is sought such that it produces a desired velocity profile. The discretisation of state and co-state variables follows a lowest-order scheme, whereas three different approaches are used for the control representation: a variational discretisation, and approximation through piecewise constant and piecewise linear elements. We employ the optimise-then-discretise approach, resulting in a non-symmetric discrete formulation. A priori error estimates for velocity, pressure, and control in natural norms are derived, and a set of numerical examples is presented to illustrate the performance of the method and to confirm the predicted accuracy of the generated approximations under various scenarios. Keywords

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Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem

Cited by 18 publications

References 46 publications

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

Error Bounds for Discontinuous Finite Volume Discretisations of Brinkman Optimal Control Problems

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