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

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