Continuous Mesh Framework Part I: Well-Posed Continuous Interpolation Error

12 years 11 months ago

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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 L1 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 well deﬁned on Riemannian metric spaces: calculus of variations, diﬀerentiation, optimization, . . . whereas these tools are not deﬁned on discrete meshes. This asset ...

Adrien Loseille, Frédéric Alauzet

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