Variational Mesh Adaptation Methods for Axisymmetrical Problems

Abstract

We study variational mesh adaptation for axially symmetric solutions to two-dimensional problems. The study is focused on the relationship between the mesh density distribution and the monitor function and is carried out for a traditional functional that includes several widely used variational methods as special cases and a recently proposed functional that allows for a weighting between mesh isotropy (or regularity) and global equidistribution of the monitor function. The main results are stated in Theorems \ref{thm4.1} and \ref{thm4.2}. For axially symmetric problems, it is natural to choose axially symmetric mesh adaptation. To this end, it is reasonable to use the monitor function in the form $G = \lambda_1(r) {\mbox{\boldmath ${e}$}}_r {\mbox{\boldmath ${e}$}}_r^T + \lambda_2(r) {\mbox{\boldmath ${e}$}} _\theta {\mbox{\boldmath ${e}$}}_\theta^T $, where ${\mbox{\boldmath ${e}$}}_r$ and ${\mbox{\boldmath ${e}$}}_\theta$ are the radial and angular unit vectors.

It is shown that when higher mesh concentration at the origin is desired, a choice of $\lambda_1$ and $\lambda_2$ satisfying $\lambda_1(0) < \lambda_2(0)$ will make the mesh denser at $r=0$ than in the surrounding area whether or not $\lambda_1$ has a maximum value at r=0. The purpose can also be served by choosing $\lambda_1$ to have a local maximum at r=0 when a Winslow-type monitor function with $\lambda_1(r) = \lambda_2(r)$ is employed. On the other hand, it is shown that the traditional functional provides little control over mesh concentration around a ring $r = r_\lambda > 0$ by choosing $\lambda_1$ and $\lambda_2$.

In contrast, numerical results show that the new functional provides better control of the mesh concentration through the monitor function. Two-dimensional numerical results are presented to support the analysis.