Sampling, approximation, and interpolation of differential forms by admissible integral k-meshes

TL;DR

The paper develops a framework for sampling, approximating, and interpolating differential forms on real bodies by extending polynomial admissible meshes to admissible integral $k$-meshes defined via averaging currents and the $0$-norm.Two constructive routes are provided: a Markov-inequality-based method that yields admissible meshes on convex bodies and a Baran-inequality-based method that reduces cardinality via the Baran distance, with explicit simplex implementations.It establishes stability and error results for least-squares fitting and interpolation of differential forms, and shows how to extract good unisolvent current sets (Fekete currents and Leja sequences) from admissible meshes, including algorithmic aspects and Lebesgue-constant bounds.Overall, the work enables robust, scalable interpolation and approximation of polynomial differential forms in the finite element exterior calculus setting, with practical strategies to control mesh size and stability.

Abstract

In this work we introduce the concept of admissible integral $k$-mesh for sampling differential forms with contiuous coefficients on a real body $E\subset \R^n$, and provide two techniques for the construction of admissible integral $k$-meshes on real bodies enjoying the Markov or the Bernstein inequality. Admissible integral $k$-meshes allow for the construction of robust approximation schemes, and are used to extract interpolation sets with high stability properties. To this end, the concepts of Fekete currents and Leja sequences of currents are formalized, and a numerical scheme for their approximation is proposed.

Figures (7)

• Figure 1: A visual representation of the construction of Proposition \ref{['prop:convmesh']} in the case $n=2$, $k=1$. The thick line represents the boundary of the convex fat set $E\subset\mathbb{R}^2$. Gray squares are the squares $Q_i$'s of the considered coordinate tessellation that intersect $E$. In this example the points $\tilde{x}_i$ and the two columns of the matrices $A^{(s,r)}$ are randomly chosen. Note that in such a way there is no control on the size of the support of the constructed functionals.
• Figure 2: Cardinalities of $\mathscr{P}_{r}\Lambda^{k}$-admissible integral $k$-meshes for the square (i.e., $n=2$) and the cube (i.e., $n=3$) constructed following Example \ref{['ex:markovsquare']} with $r$ varying from $1$ to $20$, and $k=1,\dots,n$, and $c_1=1/2$.
• Figure 3: Comparison of the cardinality of the mesh constructed in Example \ref{['ex:baransquare']} with the one constructed in Example \ref{['ex:markovsquare']} in the case of $c_1=1/2$, $n=3$, and $k=2$. As a reference we depict also the curve $r\mapsto \mathop{\mathrm{dim}}\limits \mathscr{P}_{r}\Lambda^{k}.$
• Figure 4: Elements used in Definition \ref{['def:mymesh']} in the case $n=3$. Left: the four type of simplices that are always (i.e., $k=1,2,3$) considered. Right: the tetrahedron that is taken into account only in the case $k=3$.
• Figure 5: The costruction carried out by Algorithm \ref{['alg:meshthesimplex']}. At each step the elements lyining in the grey area are not taken into account, in the next step such region is re-meshed by simplices having sides of updated length.

Theorems & Definitions (41)

• Definition 1: Integral $k$-mesh
• Remark 1
• Definition 2: (Weakly-)admissible integral $k$-mesh
• Lemma 1
• proof
• Example 1
• Theorem 1: Fundamental estimate on convex bodies
• proof
• Remark 2
• Proposition 1: Admissible integral $k$-mesh on fat convex body