A mixed finite element approximation of the Navier-Stokes equations

Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Mixed finite element approximation of a degenerate elliptic problem

Summary. We present a mixed finite element approximation of an elliptic problem with degenerate coefficients, arising in the study of the electromagnetic field in a resonant structure with cylindrical symmetry. Optimal error bounds are derived. Received May 4, 1994 / Revised version received September 27, 1994     

A practical finite element approximation of a semi-definite Neumann problem on a curved domain

Summary This paper considers the finite element approximation of the semi-definite Neumann problem: –·(u)=f in a curved domain ⁿ (n=2 or 3), on and , a given constant, for dataf andg satisfying the compatibility condition . Due to perturbation of domain errors ( ^h ) the standard Galerkin approximation to the above problem may not have a solution. A remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds. Using this approach we show that for a finite element space defined overD ^h , a union of elements, with approximation powerh ^k in theL ² norm and with dist (, ^h )Ch ^k , one obtains optimal rates of convergence in theH ¹ andL ² norms whether ^h is fitted ( ^h D ^h ) or unfitted ( ^h D ^h ) provided the numerical integration scheme has sufficient accuracy.Partially supported by the National Science Foundation, Grant #DMS-8501397, the Air Force Office of Scientific Research and the Office of Naval Research     

On the finite element approximation of a cascade flow problem

Summary The finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented. The nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements. We deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains, including the effect of numerical integration.     

Mixed finite element approximation of eddy current problems

Finite element approximations of eddy current problems thatare entirely based on the magnetic field H are haunted by theneed to enforce the algebraic constraint curl H=0 in non-conductingregions. As an alternative to techniques employing combinatorialSeifert (cutting) surfaces, in order to introduce a scalar magneticpotential we propose mixed multi-field formulations, which enforcethe constraint in the variational formulation. In light of thefact that the computation of cutting surfaces is expensive,the mixed finite element approximation is a viable option despitethe increased number of unknowns.     

Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.     

Error analysis for a finite element approximation of a thermoviscoelastic contact problem

In this article, a finite element approximation, based on a variational inequality, to the solution of a one-dimensional quasi-static Signorini contact problem in linear thermoviscoelasticity is proposed. Stability and error estimates are obtained.     

An error bound for the finite element approximation of a model for phase separation of a multi-component alloy

An error bound is proved for a fully practical piecewise linearfinite element approximation, using a backward-Euler time discretization,of a model for phase separation of a multi-component alloy.Numerical experiments with three components in one and two spacedimensions are also presented.     

A minimal model of pattern formation in a prey-predator system

The spatio-temporal dynamics of a prey-predator community is described by two reaction-diffusion equations. It is shown that for a class of initial conditions the spatio-temporal system dynamics resembles a “phase transition” between a regular and an irregular phase, separated by a moving boundary. A simple approach to specify spatio-temporal chaos is proposed.     

Polynomial approximation on tetrahedrons in the finite element method

The main aim of this paper is to derive an interpolation theorem (Theorem 1) which implies both a construction of once continuously differentiable functions which are piecewise polynomial in a domain divided into tetrahedrons (Corollary 1 and Theorem 2) and convergence theorems of the finite element method for solving three-dimensional elliptic boundary value problems of the fourth order (Theorems 3 and 4).     

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87)for analysing the time discretization error and assuming moreregularity on the initial data, we improve on the error boundderived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16,257-287) for a fully practical piecewise linear finite elementapproximation with a backward Euler time discretization of amodel for phase separation of a multi-component alloy.     

A note on the effect of numerical quadrature in finite element eigenvalue approximation

Summary In a recent work by the author and J.E. Osborn, it was shown that the finite element approximation of the eigenpairs of differential operators, when the elements of the underlying matrices are approximated by numerical quadrature, yield optimal order of convergence when the numerical quadrature satisfies a certain precision requirement. In this note we show that this requirement is indeed sharp for eigenvalue approximation. We also show that the optimal order of convergence for approximate eigenvectors can be obtained, using numerical quadrature with less precision.The author would like to thank Prof. I. Babuka for several helpful discussions. This work was done during the author's visit to the Institute of Physical Sciences and Technology and the Department of Mathematics of University of Maryland, College Park, MD 20742, USA, and was supported in part by the Office of Naval Research under Naval Research Grant N0001490-J-1030     

A finite element method for a two-sex model of population dynamics

A finite element scheme is described to approximate the solution of a nonlinear and non-local system of integro-differential equations that models the dynamics of a two-sex population. Crank-Nicolson time discretization is used and error estimates are derived for the appoximation.     

Analysis of numerical integration in p-version finite element eigenvalue approximation

We study the effect of numerical integration when the p-version of the finite element method is used to approximate the eigenpairs of elliptic partial differential operators. We obtain optimal orders of convergence for approximate eigenvalues and eigenvectors under a certain set of requirements on the quadrature rules employed. This is the same set of conditions that has been shown (in an earlier work) to be sufficient for the optimal approximation of the solutions of the corresponding source problems.     

Locking effects in the finite element approximation of elasticity problems

Summary We consider the finite element approximation of the 2D elasticity problem when the Poisson ratiov is close to 0.5. It is well-known that the performance of certain commonly used finite elements deteriorates asv0, a phenomenon calledlocking. We analyze this phenomenon and characterize the strength of the locking androbustness of varioush-version schemes using triangular and rectangular elements. We prove that thep-andh-p versions are free of locking with respect to the error in the energy norm. A generalization of our theory to the 3D problem is also discussed.The work of this author was supported in part by the Office of Naval Research under Naval Research Grant N00014-90-J-1030The work of this author was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, U.S. Air Force, under grant AFOSR 89-0252     

Finite element approximation of a nonlinear cross-diffusion population model

Summary. We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find u_i, the population of the i^th species, i=1 and 2, such that where ji and g_i(u₁,u₂):=(_i–_iiu_i–_iju_j)u_i. In the above, the given data is as follows: v is an environmental potential, c_i, a_i are diffusion coefficients, b_i are transport coefficients, _i are the intrinsic growth rates, and _ii are intra-specific, whereas _ij, ij, are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d3. Finally some numerical experiments in one space dimension are presented.Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 92D25Acknowledgements. Part of this work was carried out while the authors participated in the 2003 programme {\it Computational Challenges in Partial Differential Equations} at the Isaac Newton Institute, Cambridge, UK.