Distributional differential operators on Riemannian manifolds with smooth and Regge metrics
Date:
The slides can be found here.
Abstract:
Several finite element discretization methods rely on elements that are not globally continuous. Using appropriate duality pairings, differential operators in the sense of distributions are well-defined. They can be used to construct and analyze mixed formulations such as the Hellan–Herrmann–Johnson (HHJ), tangential-displacement normal-normal stress (TDNNS), and mass conserving mixed stress (MCS) method.
To compute curvatures of Riemannian manifolds, where the metric tensor is approximated by Regge finite elements, suitable (nonlinear) distributions are considered; see, e.g. [Gopalakrishnan J., Neunteufel M., Schöberl J. and Wardetzky M., Analysis of distributional Riemann curvature tensor in any dimension, arXiv:2311.01603, 2023]. During the numerical analysis, differential operators arise, which seem to generalize distributional differential operators from the Euclidean to the covariant setting with smooth and Regge metrics. A systemic and rigorous derivation of these operators in the sense of distributions and analysis, however, is lacking.
In this talk, we discuss the necessary tools for defining distributional covariant differential operators on Riemannian manifolds, where the metric is smooth or a Regge finite element. We focus on the covariant incompatibility operator in arbitrary dimensions. We derive several other distributional differential operators from it. Convergence rates are tested by the open-source finite element software NGSolve (www.ngsolve.org).