Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Mesh quality plays an essential role in finite element applications, since it affects the efficiency of the simulation with respect to solution accuracy and computational effort. Therefore, mesh smoothing techniques are often applied for improving mesh quality while preserving mesh topology. One of these methods is the recently proposed geometric element transformation method (GETMe), which is based on regularizing element transformations. It will be shown numerically that this smoothing method is particularly suitable, from an applicational point of view, since it leads to a significant reduction of discretization errors within the first few smoothing steps requiring only little computational effort. Furthermore, due to reduced condition numbers of the stiffness matrices the performance of iterative solvers of the resulting finite element equations is improved. This is demonstrated for the Poisson equation with a number of meshes of different complexity and type as well as for piecewise linear and quadratic finite element basis functions. Results are compared to those obtained by two variants of Laplacian smoothing and a state of the art global optimization-based approach.