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.