Section: New Results

Adaptivity with guaranteed error contraction

Participants : Martin Vohralík, Alexandre Ern, Patrik Daniel, Iain Smears.

In [26], we conceive novel adaptive refinement strategies which automatically decide between mesh refinement and polynomial degree increase. We numerically observe that the error decreases exponentially as a function of the number of degrees of freedom, for smooth as well as for singular numerical solutions. The salient feature of our approach is, however, that we ensure that the error on the next $hp$-refinement step will be reduced at least by a factor that is given. We then extend in [53] this result to the case where the underlying algebraic solver is inexact. To the best of our knowledge, these results, obtained in the framework of the Ph.D. thesis of Patrik Daniel, is the first ever where such an error contraction bound is computable and guaranteed. Numerically, its precision turns out to be very high (overestimation by a factor very close to the optimal value of one). It immediately implies convergence of the adaptive method, and we would like to use it in the near future for optimality proofs.