Abstract: We consider the discretization of differential equations satisfying the maximal parabolic $L^p$-regularity property in Banach spaces by discontinuous Galerkin (dG) methods. We use the maximal regularity framework to establish that the dG methods preserve the maximal $L^p$-regularity and satisfy corresponding a posteriori error estimates.
The a posteriori estimators are of optimal asymptotic order of convergence. A key point in our approach is a suitable interpretation of the dG methods as modified Radau IIA methods; this interpretation allows us to transfer the known maximal regularity property of Radau IIA methods to dG methods.
The talk is based on the papers:
G. A., Ch. G. Makridakis: On maximal regularity estimates for discontinuous Galerkin time-discrete methods,
SIAM J. Numer. Anal. \60 (2022) 180--194.
G. A., Ch. G. Makridakis: A posteriori error estimates for Radau IIA methods via maximal regularity.
Numer. Math. 150 (2022) 691--717.
MV:L14 and zoom