Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence (Communications in Computational Physics, 2011)
We present a novel adaptive finite element method (AFEM) for elliptic equationswhich is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergentgradient recovery. The constructions of CVT and its dual Centroidal VoronoiDelaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to producealmost equilateral two dimensional meshes. Working with finite element solutionson such high quality triangulations, superconvergent recovery methods becomeparticularly effective so that asymptotically exact a posteriori error estimations can beobtained. Through a seamless integration of these techniques, a convergent adaptiveprocedure is developed. As demonstrated by the numerical examples, the new AFEMis capable of solving a variety of model problems and has great potential in practicalapplications.