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

Anisotropic a posteriori error estimate for the virtual element method

We derive an anisotropic a posteriori error estimate for the adaptive conforming virtual element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its approximation results to prove the reliability of the error indicator. We design and implement the corresponding adaptive polygonal anisotropic algorithm. Several numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes

Mimetic finite difference method for the stokes problem on polygonal meshes

Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure

The Discrete Duality Finite Volume method for the Stokes equations on 3-D polyhedral meshes

International audienceWe develop a Discrete Duality Finite Volume (\DDFV{}) method for the three-dimensional steady Stokes problem with a variable viscosity coefficient on polyhedral meshes. Under very general assumptions on the mesh, which may admit non-convex and non-conforming polyhedrons, we prove the stability and well-posedness of the scheme. We also prove the convergence of the numerical approximation to the velocity, velocity gradient and pressure, and derive a priori estimates for the corresponding approximation error. Final numerical experiments confirm the theoretical predictions

A residual based error estimator for the Mimetic Finite Difference method

We present a local error indicator for the Mimetic Finite Difference method for diffusion-type problems on polyhedral meshes. Under essentially the same general hypotheses used in (SIAM J. Numer. Anal. 43:1872-1896, 2005) to show the convergence of the method, we prove the global reliability and local efficiency of the proposed estimator

Finite element methods with local Trefftz trial functions

In the development of numerical methods for boundary value problems, the requirement of flexible mesh handling gains more and more importance. The available work deals with a new kind of conforming finite element methods on polygonal/polyhedral meshes. The idea is to use basis functions which are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are referred to local Trefftz functions. These local problems are treated by means of boundary integral equations and are approximated by the use of the boundary element method in the numerics. The method is applied to the stationary diffusion equation, where lower as well as higher order basis functions are introduced in two space dimensions. The convergence is analysed with respect to the H^1- as well as the L_2-norm and rates of convergence are proven. In case of non-constant diffusion coefficients, a special approximation is proposed. Beside the uniform refinement, an adaptive strategy is given which makes use of the residual error estimator and an introduced refinement procedure. The reliability of the residual error estimate is proven on polygonal meshes. Finally, the generalization to arbitrary polyhedral meshes with polygonal faces is discussed. All theoretical results and considerations are confirmed by numerical experiments.In der Entwicklung numerischer Verfahren zur Approximation von Randwertaufgaben werden flexible Vernetzungen der zugrunde liegenden Gebiete immer wichtiger. Die vorliegende Arbeit beschäftigt sich mit neuartigen Finiten Element Methoden, die zu konformen Approximationen auf polygonalen und polyhedralen Gittern führen. Der Gedanke dieser Vorgehensweise liegt darin, die Ansatzfunktionen implizit als Lösungen von lokalen Randwertaufgaben zu definieren, wie dies auch schon E. Trefftz vorgeschlagen hat. Hierbei wird die Differentialgleichung des Ursprungsproblems mit konstanten Koeffizienten und homogener rechter Seite verwendet. Die lokalen Probleme werden mit Randintegralgleichungen und in der Realisierung mit Randelementmethoden behandelt. Das Verfahren wird auf die stationäre Diffusionsgleichung angewendet, wofür Ansatzfunktionen niedriger als auch höherer Ordnung eingeführt werden. Konvergenzraten bezüglich der H^1- sowie der L_2-Norm werden untersucht und bewiesen. Im Falle eines nicht konstanten Diffusionskoeffizienten wird eine spezielle Vorgehensweise vorgeschlagen. Neben der gleichmäßigen Verfeinerung der Netze wird ebenso eine adaptive Strategie angegeben, die von dem residualen Fehlerschätzer und einer eingeführten Verfeinerung Gebrauch macht. Die Zuverlässigkeit des Fehlerschätzers auf polygonalen Netzen wird bewiesen und schließlich wird das Verfahren erweitert, so dass es auf polyhedralen Gittern mit polygonalen Elementflächen angewendet werden kann. Alle theoretischen Resultate und Überlegungen werden durch numerische Experimente bestätigt