A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems

We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underling partition. We also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom. Numerical results that validate the theory are also presented

Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes

Compatible Discrete Operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that approximate constitutive relations. We devise and analyze two families of such schemes for the Stokes equations in curl formulation, with the pressure degrees of freedom located at either mesh vertices or cells. The schemes ensure local mass and momentum conservation. We prove discrete stability by establishing novel discrete Poincar\'e inequalities. Using commutators related to the consistency error, we derive error estimates with first-order convergence rates for smooth solutions. We analyze two strategies for discretizing the external load, so as to deliver tight error estimates when the external load has a large irrotational or divergence-free part. Finally, numerical results are presented on three-dimensional polyhedral meshes

Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations

Numerical results for mimetic discretization of Reissner-Mindlin plate problems

A low-order mimetic finite difference (MFD) method for Reissner-Mindlin plate problems is considered. Together with the source problem, the free vibration and the buckling problems are investigated. Full details about the scheme implementation are provided, and the numerical results on several different types of meshes are reported

Mimetic Finite Difference methods for Hamiltonian wave equations in 2D

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in time to integrate the semi-discrete Hamiltonian system. The main characteristic of MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach, associated with a symplectic method for the time integration yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper.Comment: 26 pages, 8 figure