Galerkin Least-squares Method Research Articles

A boundary integral modification of the Galerkin least squares formulation for the Stokes problem

A new boundary integral term is added to the Galerkin least squares (GLS) formulation. The new formulation compensates for a lack of consistency noted for the traditional GLS method when bilinear interpolation is used on quadrilateral elements. Exact solutions at the nodes are obtained for Poiseuille flow and another test problem, and the pressure singularity for the driven cavity is better captured.

Complex wave-number dispersion analysis of Galerkin and Galerkin least-squares methods for fluid-loaded plates

The technique of complex wave-number dispersion analysis is used to examine the accuracy of discretized systems in the representation of wave propagation phenomena. This technique was applied to study finite element approximations to in vacuo and fluid-loaded plates considering discretizations of Kirchhoff and Reissner-Mindlin plate theories coupled to an acoustic medium. The allowed evanescent waves in the finite element approximation of the in vacuo plate are obtained and compared with theory. The finite element dispersion relation for the coupled plate-fluid system is derived. Results for the Galerkin structural approximation coupled to Galerkin and Galerkin least-squares (GLS) fluid formulations are presented. Quantitative errors are reported and the significance of these errors in view of an analytical solution to the problem is given.

Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations

Symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations are presented. The Galerkin least-squares method for advective-diffusive equations is used for both systems and is related to other stabilized methods previously studied. The presentation reveals that the convergence analysis for advective-diffusive equations, as applied before to a linearized form of the compressible Navier-Stokes equations, carries over in a straightforward manner to the Stokes problem and to a linearized form of the incompressible Navier-Stokes equations.

A new finite element formulation for computational fluid dynamics: Development of an hourglass control operator for multidimensional advective-diffusive systems

An hourglass correction term is developed for multidimensional advective-diffusive systems. This term allows the use of lower order quadrature thereby making the method computationally more efficient. Numerical examples demonstrate the quality of the Galerkin least-squares method with the correction term applied to the solution of the two- and three-dimensional Navier-Stokes equations.

Generalized Galerkin Methods for Convection Dominated Transport Phenomena

A brief survey is made of recent advances in the development of finite element methods for convection dominated transport phenomena. Because of the nonsymmetric character of convection operators, the standard Galerkin formulation of the method of weighted residuals does not possess optimal approximation properties in application to problems in this class. As a result, numerical solutions are often corrupted by spurious node-to-node oscillations. For steady problems describing convection and diffusion, spurious oscillations can be precluded by the use of upwind-type finite element approximations that are constructed through a proper Petrov-Galerkin weighted residual formulation. Various upwind finite element formulations are reviewed in this paper, with a special emphasis on the major breakthroughs represented by the so-called streamline upwind Petrov-Galerkin and Galerkin least-squares methods. The second part of the paper is devoted to a review of time-accurate finite element methods recently developed for the solution of unsteady problems governed by first-order hyperbolic equations. This includes Petrov-Galerkin, Taylor-Galerkin, least-squares, and various characteristic Galerkin methods. The extension of these methods to deal with unsteady convection-diffusion problems is also considered.