Analysis of a finite volume element method for the Stokes problem

Quarteroni, Alfio; Ruiz–Baier, Ricardo

doi:10.1007/s00211-011-0373-4

Cited by 23 publications

(32 citation statements)

References 40 publications

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“…Figure 4 indicates that the method approximately attains an O(h 3/2 ) order of convergence for φ in L 2 , whereas an O(h) order is achieved for u in the H 1 seminorm and for p in L 2 . In the context of Stokes problems, with the chosen FE spaces these rates for u and p are optimal (see, e.g., [48]), and the convergence rate for φ is also in accordance with previous results for linear problems [8].…”

Section: Example 1: a Model Problem In Cartesian Coordinatessupporting

confidence: 85%

“…This paper develops, implements, and tests a numerical scheme for (1.1). In our proposed approach, the resulting system is discretized in time by a semi-implicit backward Euler method and in space by a suitable finite volume element (FVE) method whose main novelty lies in the formulation of a unified scheme for a coupled problem, while several FVE methods have been proposed for Stokes and quasi-linear scalar elliptic problems only; see, e.g., [7,15,48]. Besides the finite element (FE) primal mesh, we introduce two additional meshes on which we discretize velocity and solids fraction by continuous and discontinuous piecewise linear elements, respectively.…”

Section: Scopementioning

confidence: 99%

“…Among the possible discretizations of these variables, we restrict ourselves to continuous piecewise bilinear elements enriched with local functions for velocity (u) and continuous piecewise linear elements for the pressure (p) field (as in the multiscale stabilized method of [2,48]), whereas discontinuous piecewise linear elements are chosen for the approximation of the concentration (φ) field. In a diffusion-dominated regime, classical piecewise linear finite elements represent the most appropriate choice for the approximation of φ.…”

Section: Related Workmentioning

confidence: 99%

“…Next, the discretization of (2.2), (2.3) follows the lines of [48] in the context of the well-known variational multiscale methods [2,32]. In short, the trial functional space for u is enriched with a space of functions that do not vanish on the element boundaries and which are split into a bubble part and a harmonic extension of the local boundary condition.…”

Section: 2mentioning

confidence: 99%

“…For (3.6) it suffices to apply [19,Lemma 2.2] noting the continuity of a(ξ h )∇ a ξ h · n, whereas (3.4) and (3.5) follow as in [48].…”

Section: The Following Equations Hold For All Wmentioning

confidence: 99%

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A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

Bürger¹,

Ruiz–Baier²,

Torres³

2012

SIAM J. Sci. Comput.

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Abstract.A model of sedimentation-consolidation processes in so-called clarifier-thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the Stokes system for an incompressible fluid describing the motion of the mixture. In cylindrical coordinates, and if an axially symmetric solution is assumed, the original problem reduces to two space dimensions. This poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted Sobolev spaces. A novel finite volume element method is introduced for the spatial discretization, where the velocity field and the solids concentration are discretized on two different dual meshes. The method is based on a stabilized discontinuous Galerkin formulation for the concentration field, and a multiscale stabilized pair of P 1 -P 1 elements for velocity and pressure, respectively. Numerical experiments illustrate properties of the model and the satisfactory performance of the proposed method.

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Section: Example 1: a Model Problem In Cartesian Coordinatessupporting

confidence: 85%

Section: Scopementioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…For (3.6) it suffices to apply [19,Lemma 2.2] noting the continuity of a(ξ h )∇ a ξ h · n, whereas (3.4) and (3.5) follow as in [48].…”

Section: The Following Equations Hold For All Wmentioning

confidence: 99%

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A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

Bürger¹,

Ruiz–Baier²,

Torres³

2012

SIAM J. Sci. Comput.

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On the convergence of the Rhie–Chow stabilized Box method for the Stokes problem

Negrini,

Parolini,

Verani

2024

Numerical Methods in Fluids

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The finite volume method (FVM) is widely adopted in many different applications because of its built‐in conservation properties, its ability to deal with arbitrary mesh and its computational efficiency. In this work, we consider the Rhie–Chow stabilized Box method (RCBM) for the approximation of the Stokes problem. The Box method (BM) is a piecewise linear Petrov–Galerkin formulation on the Voronoi dual mesh of a Delaunay triangulation, whereas the Rhie–Chow (RC) stabilization is a well known stabilization technique for FVM. The first part of the article provides a variational formulation of the RC stabilization and discusses the validity of crucial properties relevant for the well‐posedness and convergence of RCBM. Moreover, a numerical exploration of the convergence properties of the method on 2D and 3D test cases is presented. The last part of the article considers the theoretically justification of the well‐posedness of RCBM and the experimentally observed convergence rates. This latter justification hinges upon suitable assumptions, whose validity is numerically explored.

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Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans

Dupraz

Filippi

Gizzi

et al. 2014

Math. Meth. Appl. Sci.

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In this paper, we are interested in the spatio-temporal dynamics of the transmembrane potential in paced isotropic and anisotropic cardiac tissues. In particular, we observe a specific precursor of cardiac arrhythmias that is the presence of alternans in the action potential duration. The underlying mathematical model consists of a reaction-diffusion system describing the propagation of the electric potential and the nonlinear interaction with ionic gating variables. Either conforming piecewise continuous finite elements or a finite volume-element scheme are employed for the spatial discretization of all fields, whereas operator splitting strategies of first and second order are used for the time integration. We also describe an efficient mechanism to compute pseudo-ECG signals, and we analyze restitution curves and alternans patterns for physiological and pathological cardiac rhythms.

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Analysis of a finite volume element method for the Stokes problem

Cited by 23 publications

References 40 publications

A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

On the convergence of the Rhie–Chow stabilized Box method for the Stokes problem

Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans

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