A Unified Divergence Analysis on Projection Method for Symm's Integral Equation of the First Kind

On bounded and simply connected planar analytic domain $ \Omega $, by $ 2\pi $ periodic regular parametric representation of boundary curve $ \partial \Omega $, complete convergence and error analysis are done in $ L^2 $ setting for least squares, dual least squares, Bubnov-Galerkin methods with trigonometric polynomials into Symm's integral equation of the first kind $ K \Psi = g $ when $ g \in H^r(0,2\pi), \ r \geq 1 $. \newline \indent In this paper, we focus on the numerical behavior of (LS), (DLS), (BG) when $ g \in H^r(0,2\pi), \ 0 \leq r < 1 $. Weakening the boundary $ \partial \Omega $ from analytic to $ C^3 $ class, it is proven that the (LS), (DLS), (BG) with trigonometric basis will uniformly diverge to infinity at first order. The divergence effect and optimality of first order rate are confirmed in an example. In particular, we show that the strong ellipticity estimate and G\"{a}rding inequality are also powerful in divergence analysis of Galerkin method on ill-posed integral equations.alerkin method on ill-posed integral equations.