A symmetric nodal conservative finite element method for the Darcy equation

This work introduces and analyzes novel stable Petrov-Galerkin EnrichedMethods (PGEM) for the Darcy problem based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where element-wise dependent residual functions, named multi-scale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multi-scale functions correct edge residuals as well. The multi-scale functions turn out to be the well-known lowest order Raviart-Thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure. The enrichment strategy suggests the way to recover the local mass conservation property for nodal-based interpolation spaces. We prove that the method and its symmetric version are well-posed and achieve optimal error estimates in natural norms. Numerical validations confirm claimed theoretical results

Stabilization arising from PGEM : a review and further developments

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated

On a residual local projection method for the Darcy equation

Abstract. A new symmetric local projection method built on residual bases (RELP) makes linear equal-order finite element pairs stable for the Darcy problem. The derivation is performed inside a Petrov-Galerkin enriching space approach (PGEM) which indicates parameter-free terms to be added to the Galerkin method without compromising consis-tency. Velocity and pressure spaces are augmented using solutions of residual dependent local Darcy problems obtained after a static condensation procedure. We prove the method achieves error optimality and indicates a way to recover a locally mass conservative velocity field. Numerical experiments validate theory. 1

Finite element pressure stabilizations for incompressible flow problems

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis

On the Limitations of Bubble Functions

We present two examples that demonstrate no advantage in enriching a finite element subspace with bubble functions. To appear in: Computer Methods in Applied Mechanics and Engineering Preprint September 1993 Acknowledgment: The authors acknowledge the support by the National Science Foundation under Grant ASC-9217394. * Visiting Associate Professor from LNCC, Rua Lauro Muller 455, 22290 Rio de Janeiro, Brazil. i L.P.Franca and C.Farhat Preprint, September 1993 1 1. Quadratics or linears? Let us consider the problem of finding the scalar valued function u(x) defined on the unit interval and satisfying \Gammau ;xx = f on (0; 1) (1) u(0) = u(1) = 0 (2) Multiplying (1) by an arbitrary function v 2 H 1 0(\Omega\Gamma --- where H 1 0(\Omega\Gamma denotes the Hilbert space of functions satisfying (2) with square integrable value and derivative on the unit interval --- and integrating on (0; 1) by parts, yields the variational formulation: Find u 2 H 1 0 (\Omega\Gamma such that a(u; ..

Bubble Functions Prompt Unusual Stabilized Finite Element Methods

A second order linear scalar differential equation including a zero-th order term is approximated using first the standard Galerkin method enriched with bubble functions. Static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations. The method is generalized to allow for a convection operator in the equation. This work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena, flows with chemical reactions, and other important problems

Unlocking with Residual-Free Bubbles

Residual-free bubbles are derived for the Timoshenko beam problem. Eliminating these bubbles the resulting formulation is form-identical to using the following tricks to the standard variational formulation: i) one-point reduced integration on the shear energy term; ii) replace its coefficient 1=ffl 2 by 1=(ffl 2 + (h 2 K =12)) in each element; iii) modify consistently the right-hand-side. This final formulation is `legally' obtained in that the Galerkin method enriched with residual-free bubbles is developed using full integration throughout. Furthermore this method is nodally exact by construction. Submitted to: Computer Methods in Applied Mechanics and Engineering Preprint July 1995 i L.P. Franca and A. Russo Preprint, July 1995 1 1. INTRODUCTION The deflection of a beam taking into account bending and shear deformations is described by the Timoshenko model. Standard Galerkin finite element method using equal-order piecewise linear approximations for the unknown dependent..