Conditional $\varepsilon$-uniform boundedness of Galerkin projectors and convergence of an adaptive mesh method as applied to singularly perturbed boundary value problems

The Galerkin finite element method is applied to nonself-adjoint singularly perturbed boundary value problems on Shishkin meshes. The Galerkin projection method is used to obtain conditionally $\varepsilon$-uniform a priori error estimates and to prove the convergence of a sequence of meshes in the case of an unknown boundary layer edge.