# Energy-corrected FEM and explicit time-stepping for parabolic problems

## TL;DR

This paper introduces an energy-corrected finite element method for parabolic problems with corners, enabling stable explicit time-stepping on quasi-uniform meshes and demonstrating effectiveness in complex geometries.

## Contribution

It develops a novel energy-corrected FEM approach that handles corner singularities without mesh refinement, allowing efficient explicit time integration in 2D and 3D.

## Key findings

- Method achieves optimal convergence despite corner singularities.
- Explicit time-stepping scheme is stable and efficient on quasi-uniform meshes.
- Numerical results confirm theoretical predictions and method flexibility.

## Abstract

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution effect". Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.