Unified analysis on Petrov-Galerkin method into Symm's integral of the first kind

TL;DR

This paper extends the convergence analysis of Petrov-Galerkin methods for Symm's integral equation from analytic to $C^3$ boundaries, and demonstrates divergence for less regular data, providing new insights into boundary regularity effects.

Contribution

It generalizes existing convergence results to less regular boundaries and analyzes divergence phenomena for low-regularity data in Petrov-Galerkin methods.

Findings

Maintains convergence analysis for $C^3$ boundaries.

Shows divergence for $g otin H^r$ with $r<1$.

Confirms divergence rate and effect through an example.

Abstract

On bounded and simply connected planar analytic domain $ \Omega $, by $ 2\pi $ periodic parametric representation of boundary curve $ \partial \Omega $, Symm's integral equation of the first kind takes form $ K \Psi = g $, where $ K $ is seen as an operator mapping from $ L^2(0,2\pi) $ to itself. The classical result show complete convergence and error analysis in $ L^2 $ setting for least squares, dual least squares, Bubnov-Galerkin methods with Fourier basis when $ g \in H^r(0,2\pi), \ r \geq 1 $. In this paper, weakening the boundary $ \partial \Omega $ from analytic to $ C^3 $ class, we maintain the convergence and error analysis from analytic case. Besides, it is proven that, when $ g \in H^r(0,2\pi), \ 0 \leq r < 1 $, the least squares, dual least squares, Bubnov-Galerkin methods with Fourier basis will uniformly diverge to infinity at first order. The divergence effect and optimality of first order rate are confirmed in an example.