GETOpt mesh smoothing: Putting GETMe in the framework of global optimization-based schemes

We present a novel approach to mesh smoothing, called GETOpt, by using a variation of the Geometric Element Transformation Method (GETMe) which falls within the framework of global optimization-based schemes and mesh quality improvement methods. We introduce a global mesh quality measure, which is inspired by rigorous convergence results concerning the size of the geometric elements (non-degeneracy and comparable sizing) and whose gradient flow yields a combination of GETMe and Laplace smoothing. Under this scheme, it can be shown rigorously that the tetrahedral mesh never degenerates. Furthermore, numerical tests show that the sizing of the mesh elements become more comparable and that the FEM gives better approximations in case the PDE problem being solved demonstrates isotropic physical behavior. More specifically, we present in this article an example where the mesh is used to solve Poisson's equation using the standard Galerkin finite element method with piecewise linear and order 2 functions on the low-order mesh.