Leopoldo P. Franca

A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of

Abstract A new Petrov-Galerkin formulation of the Stokes problem is proposed. The new formulation possesses better stability properties than the classical Galerkin/variational method. An error analysis is performed for the case in which both the velocity and pressure are approximated by C 0 interpolations. Combinations of C 0 interpolations which are unstable according to the Babuska-Brezzi condition (e.g., equal-order interpolations) are shown to be stable and convergent within the present framework. Calculations exhibiting the good behavior of the methodology are presented.

A new finite element formulation for computational fluid dynamics: VIII. The galerkin/least-squares method for advective-diffusive equations☆

Galerkin/least-squares finite-element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of streamline-upwind Petrov-Galerkin methods, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented. Some numerical results for compressible Navier-Stokes flows are presented.

A new finite formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces

Abstract Symmetric finite element formulations are proposed for the primitive-variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations. Various boundary conditions, such as pressure, are accommodated.

A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics

Results of Harten and Tadmor are generalized to the compressible Navier-Stokes equations including heat conduction effects. A symmetric form of the equations is derived in terms of entropy variables. It is shown that finite element methods based upon this form automatically satisfy the second law of thermodynamics and that stability of the discrete solution is thereby guaranteed ab initio.

Two classes of mixed finite element methods

Abstract Finite element methods are presented in an abstract settings for mixed variational formulations. The methods are constructed by adding to the classical Galerkin method various least-squares like terms. The additional terms involve integrals over element interiors, and include mesh-parameter dependent coefficients. The methods are designed to enhance stability. Consistency is achieved in the sense that exact solutions identically satisfy the variational equations. Applied to various problems, simple finite element interpolations are rendered convergent, including convenient equal-order interpolations which are generally unstable within the Galerkin approach. The methods are subdivided into two classes according to the manner in which stability is attained: 1. (1) Circumventing Babuska-Brezzi condition methods. 2. (2) Satisfying Babuska-Brezzi condition methods. Convergence is established for each class of methods. Applications of the first class of methods to Stokes flow and compressible linear elasticity are presented. The second class of methods is applied to compressible and incompressible elasticity problems.

A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems

Abstract An SUPG-type finite element method for linear symmetric multidimensional advective-diffusive systems is described and analyzed. Optimal and near optimal error estimates are obtained for the complete range of advective-diffusive behavior.

A mixed finite element formulation for Reissner—Mindlin plate theory: uniform convergence of all higher-order spaces

Abstract A new mixed finite element formulation of Reissner-Mindlin theory is presented which improves upon the stability properties of the Galerkin formulation. General convergence theorems are proved which are uniformly valid for all values of the plate thickness, including the Poisson-Kirchhoff limit. As long as the dependent variables are interpolated with functions of sufficiently high order, the formulation is convergent. No special devices are required.

A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation

SummaryAdding to the classical Hellinger Reissner formulation another residual form of the equilibrium equation, a new Petrov-Galerkin finite element method is derived. It fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations, including equal-order discontinuous stress and continuous displacement interpolations which are unstable within the Galerkin approach. Error estimates are presented using the Babuška-Brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method.

Mixed Petrov-Galerkin methods for the Timoshenko beam problem

Abstract A new mixed Petrov-Galerkin method is presented for the Timoshenko beam problem. The method has enhanced stability compared to the Galerkin formulation, allowing new combinations of interpolation, in particular, equal-order stress and displacement fields. The methodology is easily generalizable for multi-dimensional Hellinger-Reissner systems.

Petrov-Galerkin formulations of the Timoshenko beam problem

Abstract Petrov-Galerkin formulations of the Timoshenko beam problem are presented. They are shown to provide the best approximation property, optimal rate of convergence, and nodally exact solution for arbitrary loading, for all values of the thickness of the beam.