Nonlinear piecewise polynomial approximation: Theory and algorithms

We study nonlinear n-term approximation in (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of $R2$ which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel triangulations. We call them B-spaces because they can be viewed as variations of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates for n-term piecewise polynomial approximation and consequently characterize the corresponding approximation spaces by interpolation. We also develop methods for n-term piecewise polynomial approximation which provide the rate of the best approximation.; Influenced by these and further theoretical results, we develop efficient practical algorithms for compression and quick rendering of Digital Terrain Elevation Data (DTED) maps and implement these algorithms as C code.