[Show abstract][Hide abstract] ABSTRACT: We consider discretizations for reaction-diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary vari-ables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, mesh-independent global upper and lower L ∞ bounds for the chemical potentials. These bounds provide the main step for a con-vergence analysis for the full discretized nonlinear evolution problem. The funda-mental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalit-ies we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides with the gradient norm of continuous piecewise linear finite elements.
[Show abstract][Hide abstract] ABSTRACT: According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier–Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix–Raviart element is proposed, where divergence-free, lowest-order Raviart–Thomas velocity reconstructions reestablish L2L2-orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.
Computer Methods in Applied Mechanics and Engineering 01/2014; 268:782–800. · 2.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be O(1). This paper revisits this choice for the Stokes equations on the basis of minimizing the H 1 (Ω) error of the velocity and the L 2 (Ω) error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the H 1 (Ω) error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the L 2 (Ω) error of the pressure requires in many cases smaller stabilization parameters than the minimization of the H 1 (Ω) velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.
Advances in Computational Mathematics 01/2014; 40(2). · 1.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad–div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad–div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator’s matrices are identical to those of grad–div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad–div stabilization, and in all cases, solutions are found to be very similar.
Computer Methods in Applied Mechanics and Engineering 07/2013; s 261–262:142–153. · 2.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations should be balanced completely by the pressure gradient. Unfortunately, nearly all available discretizations for incompressible flows violate this property. The origin of the problem is that discrete velocities are usually not divergence-free. Hence, the use of divergence-free velocity reconstructions is proposed wherever an L2L2 scalar product appears in the discrete variational formulation. The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids. It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart–Thomas element increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.RésuméLors dʼécoulements incompressibles avec vitesses normales nulles à la frontière, les forces présentes dans les équations de conservation de la quantité de mouvement dont le rotationnel sʼannule ne doivent tre équilibrées que par le gradient de la pression. Malheureusement, cette propriété nʼest pas vérifiée par la plupart des méthodes de discrétisation disponibles, pour lesquelles la divergence (en un sens continu) de lʼapproximation de la vitesse nʼest pas nulle. Aussi, nous proposons dʼutiliser une reconstruction continue de la vitesse à divergence nulle, dans chaque produit scalaire L2L2 intervenant dans la formulation variationnelle. Nous illustrons cette méthode dans le cas dʼun schéma non conforme de type MAC sur grille non structurée de Delaunay. La reconstruction basée sur les éléments de Raviart–Thomas de bas degré, permet dʼaccroître la robustesse et la précision de ce schéma dans des cas de forces irrotationnelles significatives.
[Show abstract][Hide abstract] ABSTRACT: We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott–Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.
Computer Methods in Applied Mechanics and Engineering 09/2012; s 237–240:166–176. · 2.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions are constructed and analyzed, first on grids which satisfy an orthogonality condition, and then on general, possibly non conforming meshes. In both cases, the piece-wise constant approximate solution is shown to converge in L2 () to the exact solution; similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. These results are confirmed by numerical results.
Mathematics of Computation 01/2012; 81:2019-2048. · 1.41 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It was recently proven in Case et al. (2010)  that, under mild restrictions, grad-div stabilized Taylor–Hood solutions of Navier–Stokes problems converge to the Scott–Vogelius solution of that same problem. However, even though the analytical rate was only shown to be γ−12 (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to γ−1. We prove herein the analytical rate is indeed γ−1, and extend the result to other incompressible flow problems including Leray-α and MHD. Numerical results are given that verify the theory.
Journal of Mathematical Analysis and Applications 09/2011; 381(2):612-626. · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier–Stokes equations for the flow are discretized by the divergence-free Scott–Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection–diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.
[Show abstract][Hide abstract] ABSTRACT: This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier—Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott—Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier—Stokes equations. We also prove that the limit of the grad-div stabilized Taylor—Hood solutions to the Navier—Stokes problem converges to the Scott—Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott—Vogelius and grad-div stabilized Taylor—Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier—Stokes approximations.
[Show abstract][Hide abstract] ABSTRACT: We study extensions of the energy and helicity preserving scheme for the 3D Navier-Stokes equations, developed in (23), to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme.
International Journal of Numerical Analysis and Modeling 01/2011; · 0.67 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present numerical results for two generalized MAC schemes on triangular meshes, which are based on staggered meshes using
the Delaunay–Voronoi duality. In the first one, the pressures are defined at the vertices of the mesh, and the discrete velocities
are tangential to the edges of the triangles. In the second one, the pressures are defined in the triangles, and the discrete
velocities are normal to the edges of the triangles. In both cases, convergence results are obtained.
KeywordsNavier–Stokes-MAC scheme-Delaunay mesh
[Show abstract][Hide abstract] ABSTRACT: We show the velocity solutions to the convective, skew-symmetric, and rotational Galerkin finite element formulations of the Navier-Stokes equations are identical if Scott-Vogelius elements are used, and thus all three formulations will be the same pointwise divergence free solution velocity. A connection is then established between the formulations for grad-div stabilized Taylor-Hood elements: under mild restrictions, the formulations' velocity solutions converge to each other (and to the Scott-Vogelius solution) as the stabilization parameter tends to infinity. Thus the benefits of using Scott-Vogelius elements can be obtained with the less expensive Taylor-Hood elements, and moreover the benefits of all the formulations can be retained if the rotational formulation is used. Numerical examples are provided that confirm the theory.
Journal of Computational Physics 12/2010; 229:9020-9025. · 2.49 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In the numerical simulation of the incompressible Navier–Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete Ladyzhenskaya–Babuska–Brezzi (LBB) constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor–Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore, we present a simple steady Navier–Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed.
Computer Methods in Applied Mechanics and Engineering 09/2009; · 2.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We propose a stabilized mixed finite element method based on the Scott–Vogelius element for the Oseen equation. Here, only convection has to be stabilized since by construction both the discrete pressure and the divergence of the discrete velocities are controlled in the norm L2. As stabilization we propose either the local projection stabilization or the interior penalty stabilization based on the penalization of the gradient jumps over element edges. We prove a discrete inf–sup condition leading to optimal a priori error estimates. Moreover, convergence of the velocities is completely independent of the pressure regularity, and in the purely incompressible case the discrete velocities are pointwise divergence free. The theoretical considerations are illustrated by some numerical examples.