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

Approximation classes for adaptive time-stepping finite element methods