2025/05/25 by David Williams, Williams, David M., Mathijs Wintraecken +1
Computer Science · Mathematics · #65N15 #65N30 #65N50 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2505.18987
openalex publication_date 2025/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝd.