Construction of 2D Explicit Cubic Quasi-Interpolating Splines in Bernstein-Bézier Form

Core Concepts

The authors propose the construction of C1-cubic Hermite splines on a uniform three-direction triangulation, where the Bernstein-Bézier coefficients are determined by the values of the function and its gradient at the vertices of the triangulation. The resulting quasi-interpolation operator is exact on quadratic polynomials.

Abstract

The paper addresses the construction of C1 cubic quasi-interpolants on a three-direction mesh of R2. The quasi-interpolating splines are defined by directly setting their Bernstein-Bézier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Additional global properties are also required. The key highlights and insights are: The quasi-interpolation operator Q is defined by means of Bernstein-Bézier coefficients that are linear combinations of the function values and its first-order partial derivatives at the vertices of the triangulation. The C1 smoothness of the quasi-interpolant Q[f] is ensured by imposing a system of linear equations on the Bernstein-Bézier coefficients. The problem of finding a quasi-interpolation operator Q exact on P2 and such that Q[f] is in S1_3(Δ3) has a 5-parametric family of solutions. To achieve superconvergence at the midpoints of the edges of the triangulation, the authors provide specific values for the free parameters in the Bernstein-Bézier coefficient masks. Numerical tests are provided to confirm the approximation properties of the proposed quasi-interpolation scheme.

Stats

The quasi-interpolation error ∥f-Q*[f]∥_C1,T relative to a triangle T is of order O(h^3), where ∥f∥_C1,T := ∥f∥_∞,T + h|∂_1,0 f|_∞,T + h|∂_0,1 f|_∞,T.

Quotes

"The quasi-interpolating splines are defined by directly setting their Bernstein-Bézier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree." "The problem of finding a quasi-interpolation operator Q exact on P2 and such that Q[f] is in S1_3(Δ3) has a 5-parametric family of solutions."

Key Insights Distilled From

Construction of 2D explicit cubic quasi-interpolating splines in Bernstein-Bézier form

by Domi... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19491.pdf

Deeper Inquiries

How can the proposed quasi-interpolation scheme be extended to higher-degree splines or non-uniform triangulations

The proposed quasi-interpolation scheme can be extended to higher-degree splines by increasing the degree of the polynomials used in the Bernstein-Bézier representation. For example, instead of using cubic polynomials, one can use quartic or higher-degree polynomials to construct quasi-interpolants of higher order. This would involve defining additional Bernstein basis functions and corresponding coefficients to represent the higher-degree polynomials on each triangle of the mesh. In the case of non-uniform triangulations, the same principles can be applied by adapting the construction of the quasi-interpolants to the specific geometry of the mesh. This may involve adjusting the computation of the Bernstein basis functions and coefficients to accommodate the varying shapes and sizes of the triangles in the non-uniform triangulation. By appropriately modifying the formulation of the quasi-interpolation scheme, it can be generalized to work effectively on non-uniform meshes as well.

What are the potential applications of this quasi-interpolation method in areas such as numerical analysis, computer graphics, or scientific computing

The proposed quasi-interpolation method has various potential applications in numerical analysis, computer graphics, and scientific computing. In numerical analysis, quasi-interpolation is a valuable tool for approximating functions from limited data points, making it useful in numerical integration, function approximation, and solving differential equations. The local nature of quasi-interpolants allows for efficient and accurate approximations without the need for solving large linear systems, which is often required in traditional interpolation methods. In computer graphics, quasi-interpolation can be used for curve and surface fitting, image processing, and geometric modeling. The ability to construct smooth and continuous curves or surfaces that pass through given points with specified gradients makes quasi-interpolants suitable for generating realistic and visually appealing graphics. In scientific computing, the quasi-interpolation method can be applied to analyze and visualize complex data sets, such as in geospatial analysis, medical imaging, and computational fluid dynamics. By accurately approximating data points and gradients, quasi-interpolants can help researchers and scientists gain insights from their data and make informed decisions based on the interpolated results.

Can the superconvergence properties of the quasi-interpolant be further improved or generalized to other types of mesh structures

The superconvergence properties of the quasi-interpolant can potentially be improved by refining the choice of parameters and masks used in the construction of the quasi-interpolants. By optimizing the masks associated with the vertices and barycenters, it may be possible to achieve higher orders of convergence at specific points, not just at the midpoints of the edges. Additionally, exploring different interpolation schemes or modifying the construction method could lead to enhanced superconvergence properties in the quasi-interpolants. Generalizing the superconvergence properties to other types of mesh structures would involve adapting the construction of the quasi-interpolants to the specific characteristics of the mesh. By considering the geometry and connectivity of different mesh structures, it may be possible to derive masks and parameters that enable superconvergence at key points across various types of meshes. This generalization could enhance the accuracy and efficiency of the quasi-interpolation method in a wider range of applications and computational settings.