Functional Data Approximation on Bounded Domains using Polygonal Finite Elements.

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Computer Graphics

Background:

• Functional data approximation on 2D domains is crucial for various scientific and engineering applications.
• Traditional finite element methods often rely on simplicial meshes, which can be inefficient for complex geometries.
• Generalized barycentric coordinates offer alternative frameworks for constructing finite elements.

Purpose of the Study:

• To develop and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements.
• To evaluate the approximation quality and convergence of quadratic serendipity elements on polygonal meshes.
• To propose algorithms for generating adaptive meshes for functional data approximation.

Main Methods:

• Construction and analysis of piecewise approximations using generalized barycentric finite elements, specifically quadratic serendipity elements.
• Numerical experiments comparing approximation qualities using Wachspress, natural neighbor, Poisson, and mean value coordinates.
• Development of two greedy algorithms for generating Voronoi meshes for adaptive approximations.
• Refinement of polygonal meshes and parameter coefficients using L2-optimization.

Main Results:

• Quadratic serendipity elements on polygonal domains exhibit space/accuracy advantages over traditional finite elements on simplicial meshes.
• The proposed greedy algorithms effectively generate Voronoi meshes for adaptive functional data approximation.
• L2-optimization further enhances the piecewise functional approximation accuracy.
• Demonstrated efficacy in modeling features and discontinuities in functional data and image approximation.

Conclusions:

• Quadratic serendipity finite elements provide an efficient and accurate method for functional data approximation on polygonal domains.
• The developed greedy meshing algorithms and optimization techniques improve adaptive approximation capabilities.
• This approach shows significant potential for applications in functional data analysis and image processing.