CONTINUOUS MESH FRAMEWORK PART I: WELL-POSED CONTINUOUS INTERPOLATION ERROR

• Published in: SIAM journal on numerical analysis Vol. 49; no. 1/2; pp. 38 - 60
• Main Authors: LOSEILLE, ADRIEN, ALAUZET, FRÉDÉRIC
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Adaptation; Analysis of PDEs; Approximations and expansions; Calculus of variations and optimal control; Density; Error analysis; Errors; Euclidean space; Exact sciences and technology; Finite element method; General topology; Interpolation; Linear interpolation; Mathematical analysis; Mathematical duality; Mathematical functions; Mathematical models; Mathematics; Metric space; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Sciences and techniques of general use; Studies; Symmetry; Tensors; Tetrahedrons; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Unit ball; Vertices; 65N15; Linear interpolation; Grid pattern; Metric space; Error estimation; Optimization method; Riemannian metric space; anisotropic mesh adaptation; interpolation error; Numerical approximation; optimal interpolation error bound; Numerical analysis; linear interpolate; 65D05; 65N50; Variational calculus; 65L50; unstructured mesh; Mathematical programming; continuous mesh; optimal interpolation error bound AMS subject classifications. 65D05
• ISSN: 0036-1429; 1095-7170; 1095-7170
• DOI: 10.1137/090754078

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.