An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations

TL;DR

This paper introduces an adaptive space-time reduced basis method with certified error estimation for nonsmooth parabolic PDEs, integrating DEIM for approximation and demonstrating improved efficiency over classical methods.

Contribution

It develops a novel adaptive RB-DEIM algorithm with a certified a-posteriori error estimator tailored for nonsmooth parabolic PDEs, combining offline phases into one.

Findings

The proposed method outperforms classical RB and RB-DEIM in numerical experiments.

The error estimator effectively guides adaptivity in the reduced basis framework.

Integration of DEIM with the error estimator improves approximation of nonsmooth solutions.

Abstract

In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches.