We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral grids. To approximate the flux (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per (n-gon) element face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen inner product in the space of discrete fluxes allows for local flux elimination and reduction of the method to a symmetric cell-centered finite difference scheme for the pressure. In the case of simplicial grids, optimal first-order convergence is proved for both variables, as well as second-order convergence for the scalar variable. Numerical results confirm the theory.

Abstract. We derive a numerical method for Darcy flow, hence also for Poisson’s equation in first order form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is its discretization on simplicial complexes such as triangle and tetrahedral meshes. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. Our method requires the use of meshes in which each simplex contains its circumcenter. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus. 1.

Abstract. We present new finite element methods for Helmholtz and Maxwell equations for general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes. 1. Introduction. In

Abstract. A numerical method for the reconstruction of interfaces in finite volume schemes for multiphase flows is presented. The computation of the triple-point at the intersection of three materials in two dimensions of space is addressed. The determination of the normal vectors between pairs of materials is obtained with a finite element approximation. A numerical method for the localization of a triple-point is described as the minimum of a constrained minimization problem inside an interfacial cell of the discretization. For given volume fractions of materials in the cell, an interior-point/Newton method is used for the reconstruction of the local geometry and the localization of the triple-point. Initialization of the Newton method is performed with a derivative-free algorithm. Numerical results are presented for static and pure advection cases to illustrate the efficiency and robustness of the algorithm.

We study locally mass conservative approximations of coupled Darcy and Stokes flows on polygonal and polyhedral meshes. The discontinuous Galerkin (DG) finite element method is used in the Stokes region and the mimetic finite difference method is used in the Darcy region. DG finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces. Optimal convergence estimates for the numerical scheme are derived. Results from computational experiments supporting the theory are presented. 1

simulations of free surface flows on adaptive cartesian grids with the level set function method ∗

Most efficient adaptive mesh methods employ a few strategies, including local mesh refinement (h-adaptation), movement of mesh nodes (r-adaptation), and node reconnection (c-adaptation). Despite its simplicity, node reconnection methods are seldom analyzed apart from the other adaptation methods even in applications where severe restrictions are imposed on topological operations with a mesh. However, using only node reconnection the discretization error can be significantly reduced. In this paper, we develop and numerically analyze a new c-adaptation algorithm for mimetic finite difference discretizations of elliptic equations on triangular meshes. Our algorithm is based on a new error indicator for such discretizations, which can also be used for unstructured general polygonal meshes. We demonstrate the efficiency of our new algorithm with numerical examples.

Key words variational integrators, free-Lagrange methods, shallow water. The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by Augenbaum (1984) who discretised Hamilton’s action principle with a free-Lagrange data structure. The primary purpose of this article is to demonstrate, through the use of geometric integrators, that the VFL method exhibits no secular drift in the energy error over long-time shallow water simulations. We additionaly derive the semi-discrete divergence and potential vorticity equations in the Lagrangian frame, both of which augment the description of the discrete momentum equation by characterising the evolution of its respective irrotational and solenoidal components. Like the continuum equations, the former exhibits a div 2 U term which indicates that the flow has a very strong tendency towards a purely rotational state. The latter equation provides crucial insight into the form of discrete curl operator required for conservation of discrete potential vorticity. Numerical results demonstrate the conservative properties of the VFL method and motivate the application of the VFL method to long-time climate simulations. 1

Abstract. We consider the tensorial diffusion equation, and address the discrete maximumminimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart-Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximumminimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems. 1.