Maximum norm a posteriori error estimates for singularly perturbed
differential equations
The talk addresses the numerical solution of singularly perturbed
differential equations in one and two dimensions. Because solutions
of such problems exhibit sharp boundary and interior layers (which
are narrow regions where solutions change rapidly), a significant
economy of computer memory and time can be attained by using
special layer-adapted meshes. These meshes are fine in
layer-regions and standard outside; in two dimensions they have
extremely high maximum aspect ratios. Ideally, mesh layer
adaptation is automated by exploiting sharp a posteriori error
estimates. However, the known a posteriori error estimates are
typically under the minimum angle condition, equivalent to the
bounded-mesh-aspect-ratio condition, which is rather restrictive
and makes a posteriori error estimates less practical for layer
solutions. In contrast, we present certain new a posteriori error
estimates that hold true under no mesh aspect ratio condition.
These estimates are in the maximum norm, which is sufficiently
strong to capture layers. Furthermore, our error estimates are
uniform in the singular perturbation parameter, which is
significant since in general the error constant might blow up as
the perturbation parameter becomes small.
