Lourenço Beirão da Veiga

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.

A Higher-Order Formulation of the Mimetic Finite Difference Method

A new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes. This approach provides a higher-order accurate approximation to the flux of the exact solution. In analogy with the original formulation, a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability. The scalar solution field is approximated by a piecewise constant function. A computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable. Finally, optimal convergence rates and accuracy improvement with respect to the lower-order formulation are shown by numerical examples.

A mimetic discretization of elliptic obstacle problems

We develop a Finite Element Method (FEM) which can adopt very general meshes with polygonal elements for the numerical approximation of elliptic obstacle problems. These kinds of methods are also known as mimetic discretization schemes, which stem from the Mimetic Finite Difference (MFD) method. The first-order convergence estimate in a suitable (mesh-dependent) energy norm is established. Numerical experiments confirming the theoretical results are also presented.

H(div) and H(curl)-conforming virtual element methods.

In the present paper we construct virtual element spaces that are \(H(\mathrm{div})\)-conforming and \(H(\mathbf{curl})\)-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known finite elements. We moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations. Finally, we discuss the construction of exact sequences of VEM spaces.

A virtual element method for the acoustic vibration problem

We analyze in this paper a virtual element approximation for the acoustic vibration problem. We consider a variational formulation relying only on the fluid displacement and propose a discretization by means of

Serendipity face and edge VEM spaces

\mathrm {H}(\mathrm {div})

An Introduction to the Numerical Analysis of Isogeometric Methods

H(div) virtual elements with vanishing rotor. Under standard assumptions on the meshes, we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates. With this end, we prove approximation properties of the proposed virtual elements. We also report some numerical tests supporting our theoretical results.

Foundations of mimetic finite difference method

Abstract The asymptotic behaviour of the smallest eigenvalue in linear Koiter shell problems is studied, as the thickness parameter tends to zero. In particular, three types of shells of revolution are considered. A result concerning the ratio between the bending and the total elastic energy is also provided, by using the general theory detailed in [L. Beirao da Veiga, C. Lovadina, An interpolation theory approach to Shell eigenvalue problems (submitted for publication); L. Beirao da Veiga, C. Lovadina, Asymptotics of shell eigenvalue problems, C.R. Acad. Sci. Paris 9 (2006) 707–710].