Error estimates for the interpolation and approximation of gradients and vector fields on protected Delaunay meshes in $\mathbb{R}^d$

TL;DR

This paper derives explicit error estimates for high-order polynomial gradient interpolation and approximation on protected Delaunay meshes in multiple dimensions, extending results beyond gradients to general vector fields.

Contribution

It provides the first explicit error bounds for high-order polynomial interpolation of vector fields on protected Delaunay meshes in higher dimensions.

Findings

Error estimates depend on the minimum thickness of simplices.

Protected Delaunay meshes allow precise control of mesh quality.

Results extend to general vector fields beyond gradients.

Abstract

One frequently needs to interpolate or approximate gradients on simplicial meshes. Unfortunately, there are very few explicit mathematical results governing the interpolation or approximation of vector-valued functions on Delaunay meshes in more than two dimensions. Most of the existing results are tailored towards interpolation with piecewise linear polynomials. In contrast, interpolation with piecewise high-order polynomials is not well understood. In particular, the results in this area are sometimes difficult to immediately interpret, or to specialize to the Delaunay setting. In order to address this issue, we derive explicit error estimates for high-order, piecewise polynomial gradient interpolation and approximation on protected Delaunay meshes. In addition, we generalize our analysis beyond gradients, and obtain error estimates for sufficiently-smooth vector fields. Throughout the paper, we show that the quality of interpolation and approximation often depends (in part) on the minimum thickness of simplices in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$.