Least-Squares Finite Element Methods

JaEun Ku, Purdue University

Least-squares finite element methods have drawn considerable
attention to approximate the solutions of second-order elliptic
partial differential equations, elasticity, Stokes and Navier-Stokes
equations. In this talk, two types of least-squares methods will be
introduced. One can be considered as a stabilized Galerkin method
which uses a computable discrete negative norm.  The other is known
as a first-order least-squares method(FOLSM) based on a first-order
system representing a second-order equation. Advantages of
least-squares methods over the standard/mixed Galerkin methods will
be addressed and various new error estimates of these methods will
be presented. In case of a first-order least-squares method, we will
present separate error estimates for the primary function $u$ and
the new variable $\sigma$ which transforms a second-order equations
into a system of first-order. These separate error estimates can be
used to justify the process of using linearization step to solve
nonlinear problems.  In this regard, using least-squares solver for
Stokes equations to approximate the solutions of Navier-Stokes
equations will be discussed.