Issues Related to Least-Squares Finite Element Methods for the Stokes Equations

Abstract

Abstract. Least-squares finite element methods have become increasingly popular for the approximate solution of first-order systems of partial differential equations. Here, after a brief review of some existing theories, a number of issues connected with the use of such methods for the velocityvorticity-pressure formulation of the Stokes equations in two dimensions in realistic settings are studied through a series of computational experiments. Finite element spaces that are not covered by existing theories are considered; included in these are piecewise linear approximations for the velocity. Mixed boundary conditions, which are also not covered by existing theories, are also considered, as is enhancing mass conservation. Next, problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved. A conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh-dependent weights in the least-squares functional almost always improves the accuracy of the approximations. Concluding remarks concerning three-dimensional problems, the nonlinear Navier-Stokes equations, and the conditioning of the discrete systems are provided. Key words. least squares, finite element methods, Stokes equations AMS subject classification. 65N30PII. S10648275952945261. Introduction. Least-squares finite element methods have always held out the attraction of yielding discrete linear systems that are symmetric and positive definite even for problems for which other methods, e.g., mixed finite element methods, fail to do so; see, e.g. , [2]-[48], [50]-[56], [58], and [60]-[84]. In many settings such as the primitive variable formulation of the Stokes equations, these methods suffer from two serious problems. The first is that conforming discretizations require the use of continuously differentiable finite element functions and the second is that the condition number of the discrete equations is often proportional to h −4 , where h denotes some measure of the grid size. However, least-squares finite element methods have recently been receiving increasing attention in both the engineering and mathematics communities; see, e.g., [3] [84]. The focus of this attention has been on the application of least-squares finite element methodologies to first-order systems of partial differential equations for which one can, in principle, use merely continuous finite element functions and for which one may often prove that the condition numbers of the discrete systems are proportional to h −2 . The mathematical references cited above consider least-squares finite element methods in idealized situations, i.e., for problems having simple boundary conditions and smooth solutions and in the asymptotic limit of the grid size measure h → 0. Unfortunately, these are usually far from true in most applications of the methods to practical problems; indeed, there are many such settings for which mathematical

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

“…The least-squares formulation allows for the construction of finite element models for fluids that, when combined with high-order finite element technology [22,4,5,38,17,29,31,31,49] possess many of the attractive qualities associated with the well-known Ritz method [43] such as global minimization, best approximation with respect to a well-defined norm, and symmetric positive-definiteness of the resulting finite element coefficient matrix [9]. However, the previous applications of the least-squares method, have often been plagued with spurious solution oscillations [34] and poor conservation of physical quantities (like dilatation, mass, volume) [16]. The least-squares formulation, when combined with high-order spectral/hp finite element technology, results in a better conservation of the physical quantities and reduces the instability and spurious oscillations of solution variables with time [18,34].…”

On Mass‐Conserving Least‐Squares Methods

“…If the velocity is completely specified over all boundaries, it must be set to satisfy the global mass-conservation constraint, and only local mass conservation will not be exactly satisfied. Published computational results [16] and our own experience show that when global mass conservation is enforced through boundary conditions, LSFEM give approximate solutions with acceptable local mass conservation. However, a common flow problem involves the velocity being completely specified only along the inflow and wall regions but not completely specified along the outflow regions.…”

Least Squares Finite Element Methods

“…was studied in [28] where numerical studies showed that fairly a small weight, e.g., W = 10, helps to significantly improve total mass conservation. Thus, for the Stokes problem, at present there are methods that either recover local mass conservation but forfeit some important advantages of the RayleighRitz settings or retain all those advantages but can at best provide improved global conservation.…”