Provably size-guaranteed mesh generation with superconvergence

The mesh conditions of high-quality grids generated by bubble placement method (BPM) and their superconvergence properties are studied in this paper. A mesh condition that for each pair of adjacent triangles, the lengths of any two oppsite edges differ only by a high order of the parameter h is derived. Furthermore, superconvergence estimations are analyzed on both linear and quadratic finite elements for elliptic boundary value problems under the above mesh condition. In particular, the mesh condition is found to be applicable to many known superconvergence estimations under different types of equations. Finally, numerical examples are presented to demonstate the superconvergence properties on BPM-based grids.