A Measure of the Conformity of a Mesh to an Anisotropic Metric

A method is proposed to measure the difference between a given mesh and the size specification map that this mesh is supposed to satisfy. The size specification map is given in the form of a Riemannian metric that specifies both size and stretching of the mesh to be build. The measure evaluates the difference between the unique metric for which a simplex is a unit equilateral simplex, and the average value over the simplex of the specified Riemannian metric. Analytical and numerical examples illustrate the behavior of this measure. This measure is a unique dimensionless number that characterizes a whole mesh both in size and in shape, be it isotropic or anisotropic, coarse or fine, in a small or a big domain, in two or three dimensions.