Least-squares finite element methods

Max Gunzburger
School of Computational Science
Florida State University

Least-squares finite element methods are an attractive class of  
methods for the numerical solution of partial differential equations.  
They are motivated by the desire to recover, in general settings, the  
advantageous features of Rayleigh-Ritz methods such as the avoidance  
of discrete compatibility conditions and the production of symmetric  
and positive definite discrete systems. The methods are based on the  
minimization of convex functionals that are constructed from equation  
residuals. This paper focuses on theoretical and practical aspects of  
least-square finite element methods and includes discussions of what  
issues enter into their construction, analysis, and performance. It  
also includes a discussion of some open problems.