Grid approximation of singularly perturbed boundary value problems in a nonconvex domain with a piecewise smooth boundary

The Dirichlet problem for an elliptic equation is considered on a $L$-shaped domain formed by rectangles. The highest derivatives of the equation are multiplied by a parameter $\varepsilon$ taking arbitrary values in the half-interval (0,1]. For $\varepsilon= 0$ the elliptic equation degenerates into an equation which contains no derivatives. For the boundary-value problem, using the method on the base of classical finite difference schemes, domain decomposition and additive separation of singularities, we construct iterative and iteration-free difference schemes which do converge $\varepsilon$-uniformly. The $\varepsilon$-uniform approximation for the singular part of the solution of the boundary value problem is attained due to the use of special grids that concentrate in the neighbourhood of the boundary layer.