The multilevel finite element method for adaptive mesh optimization and visualization of volume data

Multilevel representations and mesh reduction techniques have been used for accelerating the processing and the rendering of large datasets representing scalar or vector valued functions de fined on complex 2 or 3 dimensional meshes. We present a method based on finite element approximations which combines these two ap - proaches in a new and unique way that is conceptually simple and theoretically sound. The main idea is to consider mesh reduction as an approximation problem in appropriate finite element sp aces. Starting with a very coarse triangulation of the functional domain a hierarchy of highly non-uniform tetrahedral (or triangul ar in 2D) meshes is generated adaptively by local refinement. This pro cess is driven by controlling the local error of the piecewise lin ear finite element approximation of the function on each mesh element. A reliable and efficient computation of the global approximat ion er- ror combined with a multilevel preconditioned conjugate gradient solver are the key components of the implementation. In order to analyze the properties and advantages of the adaptively generated tetrahedral meshes we implemented two volume visualization algo- rithms: an iso-surface extractor and a ray-caster. Both alg orithms, while conceptually simple, show significant speedups over c onven- tional methods delivering comparable rendering quality fr om adap- tively compressed datasets.