Algorithms for nonlinear piecewise polynomial approximation: Theoretical aspects

In this article algorithms are developed for nonlinear $n$-term Courant element approximation of functions in $L_p$ ( $0 < p \le \infty$) on bounded polygonal domains in $\mathbb{R} ^2$. Redundant collections of Courant elements, which are generated by multilevel nested triangulations allowing arbitrarily sharp angles, are investigated. Scalable algorithms are derived for nonlinear approximation which both capture the rate of the best approximation and provide the basis for numerical implementation. Simple thresholding criteria enable approximation of a target function $f$ to optimally high asymptotic rates which are determined and automatically achieved by the inherent smoothness of $f$. The algorithms provide direct approximation estimates and permit utilization of the general Jackson-Bernstein machinery to characterize $n$-term Courant element approximation in terms of a scale of smoothness spaces ($B$-spaces) which govern the approximation rates.