Nonconforming approximation methods for function reconstruction on general polygonal meshes via orthogonal polynomials

TL;DR

This paper introduces nonconforming approximation methods for function reconstruction on polygonal meshes using orthogonal polynomials, with a focus on unisolvence conditions and enrichment strategies for improved accuracy.

Contribution

It develops new nonconforming approximation techniques based on orthogonal polynomials, including unisolvence theory and enrichment methods for general polygonal meshes.

Findings

Unisolvence depends on the parity of the polynomial degree and number of edges.

Enrichment strategies can restore unisolvence when conditions are not met.

Numerical experiments confirm the accuracy of the proposed methods.

Abstract

In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials and can reproduce higher-degree polynomials. This setting naturally arises in applications where pointwise evaluations are unavailable and only integral measurements over subdomains are accessible. We develop a unisolvence theory and derive necessary and sufficient conditions for the associated approximation spaces to be unisolvent. Specifically, it is shown that unisolvence depends on the parity of the product of the polynomial degree~$m$ and the number of polygon edges~$N$. When this condition is not satisfied, we introduce an enrichment strategy involving an additional linear functional and a suitably designed enrichment function to ensure unisolvence. Numerical experiments confirm the accuracy of the proposed method.