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

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.