In this talk, we consider the numerical solution of a singularly perturbed reaction-diffusion problem:
Using a standard Galerkin method, it is natural to
conduct the analysis with respect to the
-weighted norm (energy norm),
.
This has been done in several studies, such as [2], where the
method is implemented with bilinear elements on a specially
constructed piecewise-uniform Shishkin mesh. However, the energy norm is weak for this
problem, since
and
are both
. Thus, as
,
. This is particularly problematic, since it
is easy to show that, for sufficiently small
, if the FEM
solution,
, is computed on a uniform mesh, then
is
(meaning there is no convergence, since the layers are not
resolved), yet
.
Even with the layer resolving Shishkin mesh, the error estimate in the
energy norm is
-dependent:
.
More recently, Lin and Stynes [1] have proposed the
use of a ``balanced norm'', which arises in the analysis of a mixed
finite-element method and requires
conforming
spaces. In this talk we extend these results in the framework of a
non-symmetric least-squares formulation, which finds the solution in
an
conforming finite-element space. The resulting
finite-element solutions yield
-independent
convergence in a balanced norm, resolving the boundary
layers accurately and efficiently, yet are also amenable to solution
using scalable algebraic multigrid methods.