A note on error analysis for a nonconforming discretisation of the tri-Helmholtz equation with singular data

TL;DR

This paper analyzes the error behavior of a nonconforming discretization method for the tri-Helmholtz equation with singular data, demonstrating convergence properties and regularity-based error estimates.

Contribution

It extends existing discretization analysis to the tri-Helmholtz equation with singular sources, providing new error estimates and convergence results.

Findings

Linear convergence of the discretization as mesh size decreases

Error estimates valid away from the embedded curve

Convergence aligns with classical regularity theory

Abstract

We apply the nonconforming discretisation of Wu and Xu (2019) to the tri-Helmholtz equation on the plane where the source term is a functional evaluating the test function on a one-dimensional mesh-aligned embedded curve. We present error analysis for the convergence of the discretisation and linear convergence as a function of mesh size is recovered almost everywhere away from the embedded curve which aligns with classic regularity theory.