Provably size-guaranteed mesh generation with superconvergence
For researchers in finite element methods, this work provides a theoretical foundation for achieving superconvergence on meshes with guaranteed size control, but the contribution is incremental as it extends known superconvergence theory to a specific mesh generation technique.
The paper derives a mesh condition for size-guaranteed Delaunay triangulation generated by bubble placement method (BPM) and proves superconvergence of linear and quadratic finite elements for elliptic problems under this condition. Numerical tests verify the theoretical results.
The properties and applications of superconvergence on size-guaranteed Delaunay triangulation generated by bubble placement method (BPM), are studied in this paper. First, we derive a mesh condition that the difference between the actual side length and the desired length $h$ is as small as ${\cal O}(h^{1+α})$ $(α>0)$. Second, the superconvergence estimations are analyzed on linear and quadratic finite element for elliptic boundary value problem based on the above mesh condition. In particular, the mesh condition is suitable for many known superconvergence estimations of different equations. Numerical tests are provided to verify the theoretical findings and to exhibit the superconvergence property on BPM-based grids.