A non-conditional divergence criteria of Petrov-Galerkin method for bounded linear operator equation

Petrov-Galerkin methods are always considered in numerical solutions of differential and integral equations $Ax=b$. It is common to consider the convergence and error analysis when $b \in \mathcal{R}(A)$ which make the equation solvable. However, the case when $b \notin \mathcal{R}(A)$ is always ignored. In this paper, we consider the numerical behavior of Petrov-Galerkin methods when $b \notin \mathcal{R}(A)$. It is a natural guess that when $b \in \mathcal{R}(A)$, the corresponding approximate solution constructed by Petrov-Galerkin methods with arbitrary basis will diverge to infinity. We prove this conjecture for bounded linear operator equation with dense range $\mathcal{R}(A)$ and give a more general divergence result for bounded linear operator equation with not necessarily dense range $\mathcal{R}(A)$. Several applications show its power.