First-order system least squares (FOSLS) is a fairly recent methodology aimed reformulating many PDEs as well-posed functional minimization problems. The basic idea is to begin by introducing new variables, new but consistent equations, and new boundary conditions so that the PDE becomes a well-posed first-order system. A least-squares principle is then applied, usually based on L2-type norms and possibly incorporating preconditioners and careful scaling. The aim is to produce a minimization principle that is highly accurate, easily solved, and robust. A typical successful use of FOSLS leads to a system of loosely coupled Poisson-like equations in each variable. This means that virtually any standard type of finite element discretization and multigrid solution method can obtain optimal accuracy and efficiency. Other benefits accrue as well, including an especially powerful strategy for estimating errors.
This talk is a continuation of the previous talk by Tom Manteuffel. More advanced concepts will be presented and further work will be described.
Most of our papers on FOSLS can be obtained via anonymous ftp or via the web .