Flux reconstruction and solution post-processing in mimetic finite difference methods

We present a post-processing technique for the mimetic finite difference solution of diffusion problems in mixed form. Our post-processing method yields a piecewise linear approximation of the scalar variable that is second-order accurate in the L2-norm on quite general polyhedral meshes, including non-convex and non-matching elements. The post-processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables. This technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied. The post-processing method is computationally inexpensive. Optimal performance is confirmed by numerical experiments. © 2007 Elsevier B.V. All rights reserved

Convergence analysis of the mimetic finite difference method for elliptic problems

We propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order Raviart-Thomas mixed finite element method. The theoretical results are confirmed by numerical experiments. © 2009 Society for Industrial and Applied Mathematics

Hourglass stabilization and the virtual element method

In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of C0-continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes

Virtual element method for quasilinear elliptic problems

A virtual element method for the quasilinear equation −div(κ(u)gradu)=f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the H1- and L2-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes

Virtual element methods for elliptic problems on polygonal meshes

This chapter establishes a connection between the harmonic generalized barycentric coordinates (GBCs) and the lowest order virtual element method, resulting from the fact that the discrete function space is the same for both methods. This connection allows us to look at the high order virtual element spaces as a further generalization of the harmonic GBC in both two and three spatial dimensions. We also discuss how the virtual element methodology can be used to compute approximate solutions to PDEs without requiring any evaluation of functions in the local discrete spaces, which are implicitly defined through local boundary-value problems

Hourglass stabilization and the virtual element method

In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of inline image-continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes

Virtual Element Method for Quasilinear Elliptic Problems

A Virtual Element Method (VEM) for the quasilinear equation −div(κ(u)gradu) = f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well-posedness of the discrete problem and optimal order a priori error estimates in the H1 - and L2 -norm are proven. In addition, the convergence of fixed point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.A Virtual Element Method (VEM) for the quasilinear equation −div(κ(u)gradu) = f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well-posedness of the discrete problem and optimal order a priori error estimates in the H1 - and L2 -norm are proven. In addition, the convergence of fixed point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes