Wolfgang Dahmen , Professor, Department of Mathematics, University of South Carolina
3:30 – 5PM
Tuesday Sep 11, 2018
POB 6.304
Abstract
Among the many facets of numerically solving PDEs the perhaps two core tasks are:
(i) the efficient and accurate solvability of corresponding discretized problems,
(ii) bounding the deviation of the discrete solution from the exact solution of the underlying continuous problem.
Remarkably, both tasks are traditionally treated and analyzed separately. Regarding (ii), often only a priori estimates are available that hold under sometimes unrealistic regularity assumptions. In this talk we discuss a strategy that closely intertwines (i) and (ii). In particular, it aims at producing approximate solutions that meet a given target accuracy tolerance, eventually also in scenarios where conventional schemes fail to do so. The starting point is a suitable stable variational formulation that allows one to
bound errors in the trial norm by residuals in the dual norm of an appropriate test space. The next step is to formulate an iteration in the infinite dimensional setting that
converges with a fixed error reduction per step. The numerical scheme consists then of approximately realizing this iteration within appropriately updated accuracy
tolerances so as to still guarantee convergence to the exact solution. The availability of rigorous a posteriori error estimates for the discrete sub-problems are therefore of central importance. Roughly speaking, one clings as long as possible to the infinite dimensional problem to best exploit its intrinsic metrics. We briefly sketch how this strategy can cope with the inherent obstructions encountered in two problem scenarios, namely (a) a kinetic model of radiative transfer type, and (b) the p-Poisson problem for 1