On Curved Simplicial Elements and Best Quadratic Spline Approximation for Hierarchical Data Representation

We present a method for hierarchical data approximation using curved quadratic simplicial elements for domain decomposition. Scientific data defined over two- or three-dimensional domains typically contain boundaries and discontinuities that are to be preserved and approximated well for data analysis and visualization. Curved simplicial elements make possible a better representation of curved geometry, domain boundaries, and discontinuities than simplicial elements with non-curved edges and faces. We use quadratic basis functions and compute best quadratic simplicial spline approximations that are $C^0$-continuous everywhere except where field discontinuities occur whose locations we assume to be given. We adaptively refine a simplicial approximation by identifying and bisecting simplicial elements with largest errors. It is possible to store multiple approximation levels of increasing quality. Our method can be used for hierarchical data processing and visualization.