Numerical results for mimetic discretization of Reissner–Mindlin plate problems

Abstract

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.

Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

“…We have that v T P M c P v P = v T P M P v P + v T P Γ E P v P Γ , where M P and E P are the elemental contributions to the matrices M and E, respectively. Coercivity and stability results for M c P , reported in [42], together with the quasi-uniformity assumption (19) and the definition of E, lead to…”

“…for all P ∈ Ω h and f ∈ Γ h , and where N * is a positive integer independent of h. In all the following derivations, we assume that the mesh satisfies Assumption 1, and consequently the inequalities in (19). We first introduce some norms and norm equivalence results.…”

“…Proof The equivalence relations in ( 25) is an immediate application of the definition ( 23) and ( 24), and of the inequalities in (19).…”

“…In this paper, continuing the work initiated in [12] we focus on Mimetic Finite Differences, which in the last years have been successfully applied to a wide range of problems, as for example diffusion-type problems [7,17,28,29], electromagnetism [27], plate equations [19], non-linear and control problems [8,9], and to model two-phase flows [48]. We refer to [14,18] for a comprehensive review on MFD schemes.…”

Mimetic finite differences for nonlinear and control problems

“…This is undoubtedly connected to its great flexibility in dealing with very general polygonal meshes and its capability of preserving the fundamental properties of the underlying physical and mathematical models. The MFD method has been applied with success to a wide range of linear problems, such as the diffusion problem in mixed [36][37][38][39] and primal form, 20,32 linear elasticity, 13 the Stokes equations, [17][18][19] Reissner-Mindlin plate equations, 22,24 electromagnetics, 31,33 convection-diffusion problems, 16,44 eigenvalue problems, 42 modeling of biological suspensions, 72 modeling of flows in porous media, 93 acoustic equation. 71 Issues and techniques such as satisfaction of maximum principle, 91,92 a posteriori error estimation 3,12,23 and solution post-processing 43 have been also considered.…”