Alessandro Russo

Discontinuous Galerkin Approximations for Elliptic Problems

In this article, we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven.

The Hitchhiker's Guide to the Virtual Element Method

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.

Equivalent projectors for virtual element methods

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.

Further considerations on residual-free bubbles for advective-diffusive equations

Abstract We further consider the Galerkin method for advective-diffusive equations in two dimensions. The finite dimensional space employed is of piecewise polynomials enriched with residual-free bubbles (RFB). We show that, in general, this method does not coincide with the SUPG method, unless the piecewise polynomials are spanned by linear functions. Furthermore, a simple stability analysis argument displays the effect of the RFB on the reduced space of piecewise polynomials, which, in some situations, is not equivalent to streamline diffusion for bilinears.

Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles

Abstract We show that three well-known “variational crimes” in finite elements—upwinding, mass lumping and selective reduced integration—may be derived from the Galerkin method employing the standard polynomial-based finite element spaces enriched with residual-free bubbles.

Applications of the pseudo residual-free bubbles to the stabilization of convection - diffusion problems

Abstract Residual-free bubbles have been recently introduced in order to compute optimal values for the stabilization methods a la Hughes-Franca. However, unless in very special situations (one-dimensional problems, limit cases, etc.) they require the actual solution of PDE problems (the bubble problems) in each element. Thus, they are very difficult to be used in practice. In this paper we present, for the special case of convection-dominated elliptic problems, a cheap way to compute approximately the solution of the bubble problem in each element. This provides, as a consequence, a cheap way to compute good approximations for the optimal values of the stabilization parameters.

Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations

Abstract In this paper we discuss the stabilization, via bubble functions, of a finite element method for stationary linearized incompressible Navier-Stokes equations. It is shown that ‘residual free’ bubbles can reproduce SUPG.

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

We develop an a priori error analysis of a finite element approximation to the elliptic advection-diffusion equation

New perspectives on polygonal and polyhedral finite element methods

-eps Delta u + convcdot nabla u = f

Virtual Element Method for general second-order elliptic problems on polygonal meshes

subject to a homogeneous Dirichlet boundary condition, based on the use of residual-free bubble functions. An optimal order error bound is derived in the so-called stability-norm [ biggl(eps |nabla v|^2_{L_2(Omega)} + sum_{T} h_T|convcdot nabla v|^2_{L_2(T)}biggr)^{1/2},] where hT denotes the diameter of element T in the subdivision of the computational domain.