# Space-Time Finite Element Methods for Parabolic Evolution Problems with Non-smooth Solutions

## TL;DR

This paper introduces space-time finite element methods with local stabilization and adaptive refinement for solving non-autonomous parabolic problems with low-regularity solutions, achieving improved convergence and efficient solutions.

## Contribution

It develops new stabilized finite element schemes on unstructured meshes, provides a priori estimates for low-regularity solutions, and implements adaptive refinement with efficient GMRES solvers.

## Key findings

- New a priori estimates for low-regularity solutions.
- Adaptive refinement improves convergence rates.
- Efficient GMRES solver with algebraic multigrid preconditioning.

## Abstract

We propose consistent locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of non-autonomous parabolic evolution problems under the assumption of maximal parabolic regularity. We present new a priori estimates for low-regularity solutions. In order to avoid reduced convergence rates appearing in the case of uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by algebraic multigrid.