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The principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods, and a metric tensor M and mesh metric are useful theoretical tools for understanding a methods level of mesh alignment, or anisotropy. We consider a mesh redistribution method based on the Monge-Ampere equation, which combines equidistribution of a given scalar density function with optimal transport. It does not involve explicit use of a metric tensor M, although such a tensor must exist for the method, and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh. For model problems with a linear feature and with a radially symmetric feature, we derive the exact form of the metric M, which involves expressions for its eigenvalues and eigenvectors. The eigenvectors are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues (corresponding to the level of anisotropy) is shown to depend, both locally and globally, on the value of the density function and the amount of curvature. We thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function. Numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features, including for a non-trivial time dependant nonlinear PDE which evolves narrow and curved reaction fronts., Comment: arXiv admin note: substantial text overlap with arXiv:1402.5453

Many adaptive mesh methods explicitly or implicitly use equidistribution and alignment. These principles can be considered central to mesh adaption. A Metric Tensor is the tool by which one describes the desired level of mesh anisotropy. In contrast a mesh redistribution method based on the Monge-Ampere equation, which combines equidistribution with optimal transport, does not require the explicit construction of a Metric Tensor, although one always exists. An interesting question is whether such a method produces an anisotropic mesh. To answer this question we consider the general metric to which an optimally transported mesh aligns. We derive the exact metric, involving expressions for its eigenvalues and eigenvectors, for a model linear feature. The eigenvectors of 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. 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-Ampere moving mesh method are included to verify these results, and a number of additional questions are raised.