Mixed Covolume Methods on Rectangular Grids for Elliptic Problems

We consider a covolume method for a system of first order PDEs resulting from the mixed formulation of the variable-coefficient-matrix Poisson equation with the Neumann boundary condition. The system may be used to represent the Darcy law and the mass conservation law in anisotropic porous media flow. The velocity and pressure are approximated by the lowest order Raviart-Thomas space on rectangles. The method was introduced by Russell [Rigorous Block- centered Discretizations on Irregular Grids: Improved Simulation of Complex Reservoir Systems, Reservoir Simulation Research Corporation, Denver, CO, 1995] as a control-volume mixed method and has been extensively tested by Jones [A Mixed Finite Volume Elementary Method for Accurate Computation of Fluid Velocities in Porous Media, University of Colorado at Denver, 1995] and Cai et al. [Computational Geosciences, 1 (1997), pp. 289-345]. We reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures

Optimal Order Convergence Implies Numerical Smoothness

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability

Mixed Upwinding Covolume Methods on Rectangular Grids for Convection-diffusion Problems

We consider an upwinding covolume or control-volume method for a system of rst order PDEs resulting from the mixed formulation of a convection-di usion equation with a variable anisotropic di usion tensor. The system can be used to model the steady state of the transport of a contaminant carried by a °ow. We use the lowest order Raviart{Thomas space and show that the concentration and concentration °ux both converge at one-half order provided that the exact °ux is in H1(­)2 and the exact concentration is in H1(­). Some numerical experiments illustrating the error behavior of the scheme are provided

Lorentzian area measures and the Christoffel problem

We introduce a particular class of unbounded closed convex sets of $\R^{d+1}$, called F-convex sets (F stands for future). To define them, we use the Minkowski bilinear form of signature $(+,...,+,-)$ instead of the usual scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic space \H^d. Important examples are embeddings of the universal cover of so-called globally hyperbolic maximal flat Lorentzian manifolds. Basic tools are first derived, similarly to the classical study of convex bodies. For example, F-convex sets are determined by their support function, which is defined on \H^d. Then the area measures of order $i$, $0\leq i\leq d$ are defined. As in the convex bodies case, they are the coefficients of the polynomial in $\epsilon$ which is the volume of an $\epsilon$ approximation of the convex set. Here the area measures are defined with respect to the Lorentzian structure. Then we focus on the area measure of order one. Finding necessary and sufficient conditions for a measure (here on \H^d) to be the first area measure of a F-convex set is the Christoffel Problem. We derive many results about this problem. If we restrict to "Fuchsian" F-convex set (those who are invariant under linear isometries acting cocompactly on \H^d), then the problem is totally solved, analogously to the case of convex bodies. In this case the measure can be given on a compact hyperbolic manifold. Particular attention is given on the smooth and polyhedral cases. In those cases, the Christoffel problem is equivalent to prescribing the mean radius of curvature and the edge lengths respectively

The limit set of subgroups of a class of arithmetic groups

While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider nonelementary subgroups of arithmetic groups in PSL(2,C)^q\timesPSL(2,R)^r with $q+r>1$ and their limit set. We give a necessary and sufficient condition for the projective limit set to be a point. Furthermore, we study the topology of the whole limit set

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid approximation) measured in suitable norms are derived, showing optimal orders of convergence

The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem

AbstractA finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal H1 error estimate of the finite volume solution uh and L2 error estimate for ph are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM

Error bounds for discrete minimizers of the Ginzburg-Landau energy in the high-κ\kappa regime

In this work, we study discrete minimizers of the Ginzburg-Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg-Landau parameter $\kappa$. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of $\kappa$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $\kappa$ and the spatial mesh width $h$. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted $\kappa$-weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we further explore the asymptotic optimality of our derived $L^2$- and $H^1$-error estimates with respect to $\kappa$ and $h$. Preasymptotic effects are observed for large mesh sizes $h$