Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 49, Issue 1, Pages 38-60Publisher
SIAM PUBLICATIONS
DOI: 10.1137/090754078
Keywords
unstructured mesh; continuous mesh; Riemannian metric space; interpolation error; linear interpolate; anisotropic mesh adaptation; optimal interpolation error bound
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In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density, and stretching of anisotropic meshes. But, such structures are only considered to compute lengths in adaptive mesh generators. In this article, a Riemannian metric space is shown to be more than a way to compute a length. It is proven to be a reliable continuous mesh model. In particular, we demonstrate that the linear interpolation error can be evaluated continuously on a Riemannian metric space. From one hand, this new continuous framework proves that prescribing a Riemannian metric field is equivalent to the local control in the L(1) norm of the interpolation error. This proves the consistency of classical metric-based mesh adaptation procedures. On the other hand, powerful mathematical tools are available and are well defined on Riemannian metric spaces: calculus of variations, differentiation, optimization, ... , whereas these tools are not defined on discrete meshes.
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