Approximation classes for adaptive time-stepping finite element methods

TL;DR

This paper develops approximation classes for adaptive time-stepping finite element methods applied to time-dependent PDEs, introducing new Besov space embeddings and error estimates for discontinuous in time and continuous in space finite element approximations.

Contribution

It defines novel approximation classes for adaptive time-stepping FEM, including Besov space characterizations and error estimates for mixed finite element discretizations.

Findings

Established approximation classes for adaptive FEM in time-dependent PDEs

Derived Besov space embeddings for vector-valued functions

Provided Jackson- and Whitney-type estimates for error analysis

Abstract

We study approximation classes for adaptive time-stepping finite element methods for time-dependent Partial Differential Equations (PDE). We measure the approximation error in $L_2([0,T)\times\Omega)$ and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a byproduct we define Besov spaces for vector-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates.