The method of total approximation for singularly perturbed elliptic equations with convective terms

The Dirichlet problem for elliptic equations is considered on an $n$-dimensional parallelepiped. The highest derivatives of the equation are multiplied by a parameter $\varepsilon$ taking arbitrary values from the half-interval (0,1]. When $\varepsilon=0$, the elliptic equations degenerate into first-order ones which contain derivatives with respect to the space variables, i.e. convective terms. To solve the boundary value problem, we construct a finite difference scheme that converges $\varepsilon$-uniformly. The construction of this scheme is done on the basis of the method of total approximation; $\varepsilon$-uniform convergence of the difference scheme is achieved due to the use of special piecewise uniform meshes condensing in the neighbourhood of boundary layers.