THE ALIGNMENT PROPERTIES OF MONGE-AMPÈRE BASED MESH REDISTRIBUTION METHODS: I LINEAR FEATURES

Abstract. Many adaptive mesh methods explicitly or implicitly use equidistribution and align-ment. These principles can be considered central to mesh adaption [25]. A Metric Tensor M is the tool by which one describes the desired level of mesh anisotropy. In contrast a mesh redistribution method based on the Monge-Ampère equation [9], [10], [8], [7], which combines equidistribution with optimal transport, does not require the explicit construction of a Metric Tensor M, although such an M always exists. An interesting question is whether such a method produces an anisotropic mesh. To answer this question we consider the general metric M to which an optimally transported mesh aligns. We derive the exact metric M, involving expressions for its eigenvalues and eigenvectors, for a model linear feature. The eigenvectors of M are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues is shown to depend, both locally and globally, on the value of the scalar density function ρ = det M. We thereby demonstrate how an optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function. Numerical results for a Parabolic Monge-Ampère moving mesh method [9], [10], [8], [7], [6] are included to verify these results, and a number of additional questions are raised