Ultra-weak formulation of a hypersingular integral equation on polygons and DPG method with optimal test functions

TL;DR

This paper introduces an ultra-weak formulation for hypersingular integral equations on polygons and develops a DPG method with optimal test functions, demonstrating quasi-optimal convergence through theoretical analysis and numerical experiments.

Contribution

It proposes a novel ultra-weak formulation for hypersingular equations on polygons and a DPG method with proven convergence properties.

Findings

Proved well-posedness and equivalence of the ultra-weak formulation.

Established quasi-optimal convergence of the DPG method in L2.

Validated theoretical results with numerical experiments on open curves.

Abstract

We present an ultra-weak formulation of a hypersingular integral equation on closed polygons and prove its well-posedness and equivalence with the standard variational formulation. Based on this ultra-weak formulation we present a discontinuous Petrov-Galerkin method with optimal test functions and prove its quasi-optimal convergence in $L^2$. Theoretical results are confirmed by numerical experiments on an open curve with uniform and adaptively refined meshes.