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

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 is illustrated by deriving an optimal interpolation error bound in Lp norm using calculus of variations. This result is then used as a basis to drive an anisotropic mesh refinement process.