Least-Squares Methods for Partial Differential Equations

Steve McCormick

Department of Applied Mathematics
Campus Box 526
University of Colorado at Boulder
Boulder, CO 80309-0526


Least-squares methods have become increasingly popular for solving a wide variety of partial differential equations (PDEs). First-order system least squares (FOSLS) is a special type of least-squares method that attempts to reformulate the PDE so that it is self-adjoint with an associated energy functional that is product H1 equivalent. Some of the compelling features of the FOSLS methodology include:

* self-adjoint equations, stemming from the minimization principle;
* good operator 'conditioning', stemming from the use of first-order formulations of the PDE or inverse norms in the minimization principle;
* finite element and multigrid performance that is optimal and uniform in certain parameters (e.g., Reynolds number, Poisson ratio, and wave number), stemming from uniform product norm equivalence results.

Unfortunately, one of its limitations is that these benefits come only when the original problem exhibits sufficient smoothness (it typically requires H2 regularity). An alternative to the usual L2 form of FOSLS is to use inverse norms, but this is generally very awkward and it comes with substantially reduced efficiency.

This talk will describe an alternative, called FOSLL*, that exhibits the generality of the inverse-norm FOSLS, but retains the efficiency and simplicity of the L2-norm FOSLS.