# $hp$-FEM for the fractional heat equation

## TL;DR

This paper develops an $hp$-Finite Element method combined with a Caffarelli-Silvestre extension and $hp$-Discontinuous Galerkin timestepping to solve the fractional heat equation, proving exponential convergence.

## Contribution

It introduces a novel discretization scheme for the fractional heat equation using $hp$-FEM and DG timestepping, with a rigorous convergence proof.

## Key findings

- Exponential convergence of the proposed method.
- Effective discretization of nonlocal fractional operators.
- Rigorous theoretical framework for the scheme.

## Abstract

We consider a time dependent problem generated by a nonlocal operator in space. Applying a discretization scheme based on $hp$-Finite Elements and a Caffarelli-Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using $hp$-Discontinuous Galerkin based timestepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the original domain $\Omega$.