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

Abstract

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

3D Parallel Anisotropic Unsteady Mesh Adaptation for the High-Fidelity Prediction of Bubble Motion

“…Anisotropic mesh adaptation, however, must control the sizes along prescribed directions. To this end, we use the unit-mesh concept [13] in the continuous mesh framework [14]. The main idea is to generate a uniform mesh with respect to a Riemannian metric space rather than to the Euclidian space.…”

“…where |H ρ | is derived from H ρ by taking the absolute value of the eigenvalues and π M ρ is the continuous interpolate defined in [14] and [15]. Minimizing ρ − Π h ρ L p (Ω) for a given number N of vertices can be recast in the continuous setting as minimizing ρ − π M ρ L p (Ω) for a complexity C(M) = N , which is the continuous counter part of the number of vertices.…”

Adaptive metric-based multigrid for a Poisson problem with discontinuous coefficients,

“…We propose to work in the continuous mesh framework introduced in [10,11]. The main idea of this framework is to model discrete meshes by Riemannian metric fields.…”

“…Given a continuous mesh M , we shall say, following [10,11], that a discrete mesh H of the same domain Ω is a unit mesh with respect to M , if each triangle K ∈ H , defined by its list of edges (e i ) i=1...3 , verifies:…”

“…Given a smooth function u, to each unit mesh H with respect to M corresponds a local interpolation error |u − Π H u|. In [10,11], it is shown that all these interpolation errors are well represented by the so-called continuous interpolation error related to M , which is locally expressed in terms of the Hessian H u of u as follows:…”