Adaptive Tree Approximation with Finite Element Functions: A First Look

We provide an introduction to adaptive tree approximation with finite element functions over meshes that are generated by bisection. This approximation technique can be seen as a benchmark for adaptive finite element methods, but may be also used therein for the approximation of data and coarsening. Correspondingly, we focus on approximation problems related to adaptive finite element methods, the design and performance of algorithms, and the resulting convergence rates, together with the involved regularity. For simplicity and clarity, these issues are presented and discussed in detail in the univariate case. The additional technicalities and difficulties of the multivariate case are briefly outlined.