Gabriel R. Barrenechea

An unusual stabilized finite element method for a generalized Stokes problem

Summary. An unusual stabilized finite element is presented and analyzed herein for a generalized Stokes problem with a dominating zeroth order term. The method consists in subtracting a mesh dependent term from the formulation without compromising consistency. The design of this mesh dependent term, as well as the stabilization parameter involved, are suggested by bubble condensation. Stability is proven for any combination of velocity and pressure spaces, under the hypotheses of continuity for the pressure space. Optimal order error estimates are derived for the velocity and the pressure, using the standard norms for these unknowns. Numerical experiments confirming these theoretical results, and comparisons with previous methods are presented.

Stabilized Finite Element Methods Based on Multiscale Enrichment for the Stokes Problem

This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov--Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods.

Consistent Local Projection Stabilized Finite Element Methods

This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov-Galerkin strategy for the Stokes problem. The initial unstable

Convergence Analysis of a Residual Local Projection Finite Element Method for the Navier-Stokes Equations

\mathbb{P}_1\times\mathbb{P}_l

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

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A local projection stabilized method for fictitious domains

l=0,1

Edge-based nonlinear diffusion for finite element approximations of convection---diffusion equations and its relation to algebraic flux-correction schemes

, finite element space is enhanced with solutions of residual-based local problems, and then the static condensation procedure is applied to derive new methods. The approach keeps degrees of freedom unchanged while giving rise to new stable and consistent methods for continuous and discontinuous approximation spaces for the pressure. The resulting methods do not need the use of a macroelement grid structure and are parameter-free. The numerical analysis is carried out showing optimal convergence in natural norms, and moreover, two ways of rendering the velocity field locally mass conservative are proposed. Some numerics validate the theoretical results.

On the coupling of boundary integral and finite element methods with nonlinear transmission conditions

This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs

Finite element eigenvalue enclosures for the Maxwell operator

\mathbb{P}_1/\mathbb{P}_l

A Symmetric Nodal Conservative Finite Element Method for the Darcy Equation

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