Interpolation error estimates for mean value coordinates over convex polygons

Abstract

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, to appear], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradient of the mean value coordinates does not become large as interior angles of the polygon approach π.

Bijective Composite Mean Value Mappings

“…We admit that the guaranteed bijectivity of composite mean value mappings depends on the existence of the upper bound M * in (6), which is proven only in the convex setting up to now [RGB12]. It remains future work to establish similar bounds for the gradients of mean value coordinates with respect to arbitrary polygons, but all our numerical tests show that such a bound exists and that bijective mappings can be designed by using a sufficiently large number of steps.…”

“…Another advantage of Sederberg et al's method is that it preserves convexity by construction. Thus, if source and target polygon are both convex, it is possible to derive an upper bound on M * in (6) explicitly by following the considerations in [RGB12, Theorem 4.3], and hence deduce a partition τ that guarantees the bijectivity of f τ . We carried out these calculations for the example in Figure 2 and found that a uniform composite mean value mapping with m = 10 steps is provably bijective, although we admit that it is probably preferable to simply use a bijective Wachspress mapping in this example.…”

On the construction of general cubature formula by flat extensions

“…Applications of such algorithms determining cubature rules and cubature points over general domains occur in isogeometric modeling and finite element analysis using generalized Barycentric finite elements [17,1,35,36]. Additional applications abound in numerical integration for low dimensional (6–100 dimensions) convolution integrals that appear naturally in computational molecular biology [3,2], as well in truly high dimensional (tens of thousands of dimensions) integrals that occur in finance [32,8].…”

Divergence free virtual elements for the stokes problem on polygonal meshes

“…By construction (see (32), (34) and (35)) the discrete bilinear form a h (·, ·) is (uniformly) stable with respect to the V norm. Therefore, the existence and the uniqueness of the solution to Problem (39) will follow if a suitable inf-sup condition is fulfilled, which is the topic of Section 4.1.…”

“…We refer to the recent papers and monographs [19,8,17,9,13,15,28,30,31,33,35,34,36,39,40,24,32,21] as a brief representative sample of the increasing list of technologies that make use of polygonal/polyhedral meshes. We mention here in particular the polygonal finite elements, that generalize finite elements to polygons/polyhedrons by making use of generalized non-polynomial shape functions, and the mimetic discretisation schemes, that combine ideas from the finite difference and finite element methods.…”