L. D. Marini

Two families of mixed finite elements for second order elliptic problems

SummaryTwo families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL2(Ω) andH−5(Ω) are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.

VIRTUAL ELEMENTS FOR LINEAR ELASTICITY PROBLEMS

We discuss the application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case. Virtual elements are very close to mimetic finite differences (see, for linear elasticity, [L. Beirao da Veiga, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 231--250]) and in particular to higher order mimetic finite differences. As such, they share the good features of being able to represent in an exact way certain physical properties (conservation, incompressibility, etc.) and of being applicable in very general geometries. The advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity.

A relaxation procedure for domain decomposition methods using finite elements

SummaryWe present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations.

An Inexpensive Method for the Evaluation of the Solution of the Lowest Order Raviart–Thomas Mixed Method

For the model problem

The Hitchhiker's Guide to the Virtual Element Method

- {\operatorname{div}}(a\nabla u) = f

Equivalent projectors for virtual element methods

, we show how the approximate solution produced by the Raviart–Thomas [Lecture Notes in Mathematics 606, Springer-Verlag, Berlin, 1977] mixed method (of lowest degree) can be obtained from the solution of the

Two-dimensional exponential fitting and applications to drift-diffusion models

P_1

DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS

nonconforming Galerkin method modified in a virtually cost-free manner.

A Priori Error Analysis of Residual-Free Bubbles for Advection-Diffusion Problems

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

Numerical simulation of semiconductor devices

In the original virtual element space with degree of accuracy k, projector operators in the H^1-seminorm onto polynomials of degree @?k can be easily computed. On the other hand, projections in the L^2 norm are available only on polynomials of degree @?k-2 (directly from the degrees of freedom). Here, we present a variant of the virtual element method that allows the exact computations of the L^2 projections on all polynomials of degree @?k. The interest of this construction is illustrated with some simple examples, including the construction of three-dimensional virtual elements, the treatment of lower-order terms, the treatment of the right-hand side, and the L^2 error estimates.