Variational approximation of flux in conforming finite element methods for elliptic partial differential equations: a model problem

We consider the approximation of elliptic boundary value problems by conforming finite element methods. A model problem, the Poisson equation with Dirichlet boundary conditions, is used to examine the convergence behavior of flux defined on an internal boundary which splits the domain in two. A variational definition of flux, designed to satisfy local conservation laws, is shown to lead to improved rates of convergence

Discontinuous Galerkin methods for first-order hyperbolic problems

In this paper we consider discontinuous Galerkin (DG) finite element approximations of a model scalar linear hyperbolic equation. We show that in order to ensure continuous stabilization of the method it suffices to add a jump-penalty-term to the discretized equation. In particular, the method does not require upwinding in the usual sense. For a specific value of the penalty parameter we recover the classical discontinuous Galerkin method with upwind numerical flux function. More generally, using discontinuous piecewise polynomials of degree $k$, the familiar optimal $\mathcal{O}(h^{k+1/2})$ error estimate is proved for any value of the penalty parameter. As precisely the same jump -term is used for the purposes of stabilizing DG approximations of advection-diffusion operators, the discretization proposed here can simplify the construction of discontinuous Galerkin finite element approximations of advection-diffusion problems. Moreover, the use of the jump-stabilization makes the analysis simpler and more elegant

Residual-free bubbles for advection-diffusion problems: the general error analysis

We develop the general a priori error analysis of residual-free bubble finite element approximations to linear elliptic convection-dominated diffusion problems subject to homogeneous Dirichlet boundary condition. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree greater than or equal to 1

A simple preconditioner for a discontinuous Galerkin method for the Stokes problem

In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is H(div)-conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system.Comment: 27 pages, 4 figure

Modeling Subgrid Viscosity for Advection--Diffusion Problems

We analyse the effect of the subgrid viscosity on a finite element discretisation, with piecewise linear elements, of a linear advection-diffusion scalar equation. We point out the importance of a proper tune-up of the viscosity coefficient, and we propose a heuristic method for obtaining reasonable values for it. The extension to more general problems is then hinted in the last section

Applications of nonvariational finite element methods to Monge--Amp\`ere type equations

The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge--Amp\`ere type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature.Comment: 7 pages, 3 figures, tech repor

Serendipity Nodal VEM spaces

Abstract We introduce a new variant of Nodal Virtual Element spaces that mimics the "Serendipity Finite Element Methods" (whose most popular example is the 8-node quadrilateral) and allows to reduce (often in a significant way) the number of internal degrees of freedom. When applied to the faces of a three-dimensional decomposition, this allows a reduction in the number of face degrees of freedom: an improvement that cannot be achieved by a sim- Finite Elements, but are much more robust with respect to element distortions. On more general polytopes the Serendipity VEMs are the natural (and simple) generalization of the simplicial case