Gianmarco Manzini

Discontinuous Galerkin Approximations for Elliptic Problems

In this article, we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven.

The mimetic finite difference method for elliptic problems

1 Model elliptic problems.- 2 Foundations of mimetic finite difference method.- 3 Mimetic inner products and reconstruction operators.- 4 Mimetic discretization of bilinear forms.- 5 The diffusion problem in mixed form.- 6 The diffusion problem in primal form.- 7 Maxwells equations. 8. The Stokes problem. 9 Elasticity and plates.- 10 Other linear and nonlinear mimetic schemes.- 11 Analysis of parameters and maximum principles.- 12 Diffusion problem on generalized polyhedral meshes.

Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems

The maximum principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the discrete maximum principle (DMP), is one of the most difficult properties to achieve in numerical methods, especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic. Violation of the DMP may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one. In this work, we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference (MFD) method on two- and three-dimensional meshes. These conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element. Efficient solutions are devised for meshes consisting of simplexes, parallelograms and parallelepipeds, and orthogonal locally refined elements as those used in the AMR methodology. On simplicial meshes, it turns out that the MFD method coincides with the mixed-hybrid finite element methods based on the low-order Raviart-Thomas vector space. Thus, in this case we recover the well-established conventional angle conditions of such approximations. Instead, in the other cases a suitable design of the MFD method allows us to formulate a monotone discretization for which the existence of a DMP can be theoretically proved. Moreover, on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the Multi-Point Flux Approximation (MPFA) methods. Numerical experiments confirm the effectiveness of the considered monotonicity conditions.

Convergence analysis of the high-order mimetic finite difference method

We prove second-order convergence of the conservative variable and its flux in the high-order MFD method. The convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients. For the case of non-constant coefficients, we also develop a new family of high-order MFD methods. Theoretical result are confirmed through numerical experiments.

The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes

We extend the mimetic finite difference (MFD) method to the numerical treatment of magnetostatic fields problems in mixed div-curl form for the divergence-free magnetic vector potential. To accomplish this task, we introduce three sets of degrees of freedom that are attached to the vertices, the edges, and the faces of the mesh, and two discrete operators mimicking the curl and the gradient operator of the differential setting. Then, we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div-curl variational formulation of the magnetostatic equations. This construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the MFD method. We also discuss the linear algebraic form of the resulting MFD scheme, its practical implementation, and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by Hyman and Shashkov to the case of unstructured polyhedral meshes. The accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem.

A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation

Abstract We present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes. These mimetic discretizations are built to satisfy local consistency and stability conditions. The consistency condition is an exactness property, i.e., the mimetic schemes are exact when the solution is a polynomial of an assigned degree. The stability condition ensures the well-posedness of the method. The degrees of freedom are the solution moments on mesh faces and inside mesh cells. Higher order schemes are built using higher order moments. The developed schemes are verified numerically on diffusion problems with constant and spatially variable (possibly, discontinuous) tensorial coefficients.

Mimetic scalar products of discrete differential forms

We propose a strategy for the systematic construction of the mimetic inner products on cochain spaces for the numerical approximation of partial differential equations on unstructured polygonal and polyhedral meshes. The mimetic inner products are locally built in a recursive way on each k-cell and, then, globally assembled. This strategy is similar to the implementation of the finite element methods. The effectiveness of this approach is documented by deriving mimetic discretizations and testing their behavior on a set of problems related to the Maxwell equations.

Bad behavior of Godunov mixed methods for strongly anisotropic advection-dispersion equations

We study the performance of Godunov mixed methods, which combine a mixed-hybrid finite element solver and a Godunov-like shock-capturing solver, for the numerical treatment of the advection-dispersion equation with strong anisotropic tensor coefficients. It turns out that a mesh locking phenomenon may cause ill-conditioning and reduce the accuracy of the numerical approximation especially on coarse meshes. This problem may be partially alleviated by substituting the mixed-hybrid finite element solver used in the discretization of the dispersive (diffusive) term with a linear Galerkin finite element solver, which does not display such a strong ill conditioning. To illustrate the different mechanisms that come into play, we investigate the spectral properties of such numerical discretizations when applied to a strongly anisotropic diffusive term on a small regular mesh. A thorough comparison of the stiffness matrix eigenvalues reveals that the accuracy loss of the Godunov mixed method is a structural feature of the mixed-hybrid method. In fact, the varied response of the two methods is due to the different way the smallest and largest eigenvalues of the dispersion (diffusion) tensor influence the diagonal and off-diagonal terms of the final stiffness matrix. One and two dimensional test cases support our findings.

Monotonicity Conditions in the Mimetic Finite Difference Method

The maximum principle is a major property of solutions of partial differential equations. In this work, we analyze a few constructive algorithms that allow one to embed this property into a mimetic finite difference (MFD) method. The algorithms search in the parametric family of MFD methods for a member that guarantees the discrete maximum principle (DMP). A set of sufficient conditions for the DMP is derived for a few types of meshes. For general meshes, a numerical optimization procedure is proposed and studied numerically.

Benchmark 3D: CeVeFE-DDFV, a discrete duality scheme with cell/vertex/face+edge unknowns

The method that we investigate in this contribution was proposed by Y. Coudiere and F. Hubert in [1] as a three-dimensional (3D) extension of the finite volume scheme previously studied by F. Hermeline in [4] and K. Domelevo and P. Omn`es in [3]. This method belongs to the family of Discrete Duality Finite Volume (DDFV) methods, which can naturally handle anisotropic or non-linear problems on general distorted meshes.