In this paper boundary value problems are studied for systems with large parameter, elliptic in the sense of Douglis--Nirenberg. We restrict ourselves on model problems acting in the half-space. It is possible to define parameter-ellipticity for such problems, in particular we formulate Shapiro--Lopatinskii type conditions on the boundary operators. It can be shown that parameter-elliptic boundary value problems are uniquely solvable and that their solutions satisfy uniform a priori estimates in parameter-dependent norms. We essentially use ideas from Newton's polygon method and of Vishik--Lyusternik boundary layer theory.

Mathematical problems and theory of numerical methods