Exponential Convergence of $hp$-Time-Stepping in Space-Time Discretizations of Parabolic PDEs

TL;DR

This paper proves exponential convergence of an $hp$-time-stepping method for parabolic PDEs, leveraging solution analyticity in time and combining it with spatial finite element refinements for efficient space-time discretizations.

Contribution

It establishes the exponential convergence of a conforming semi-discrete $hp$-time-stepping approach combined with spatial finite element refinements for parabolic PDEs in polygonal domains.

Findings

Exponential convergence rate of the $hp$-time-stepping method.

Error behavior comparable to standard FEM for elliptic problems.

Analysis focused on two-dimensional spatial domains.

Abstract

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this semi-discretization in time with first-order, so-called "$h$-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of $D$, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in $D$. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.