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

On bounded and simply connected planar analytic domain $\mathrm{\Omega }$, by $2\pi$ periodic parametric representation of boundary curve $\mathrm{\partial }\mathrm{\Omega }$, Symm’s integral equation of the first kind takes form $K\mathrm{\Psi }=g$, 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 $g\in H^r(0,2\pi ),r\ge 1$. In this paper, weakening the boundary $\mathrm{\partial }\mathrm{\Omega }$ from analytic to $C^3$ class, we maintain the convergence and error analysis from analytic case. Besides, it is proven that, when $g\in H^r(0,2\pi ),0\le r<1$, 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.