Regularity Estimates up to the Boundary for Elliptic Systems of Difference Equations

This paper proves regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations. The regularity estimates, obtained for boundary-fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudodifference operators to treat systems with variable coefficients, The elliptic systems of difference equations and the boundary conditions that are considered are very general in form. It is proved that regularity of a regular elliptic system of difference equations is equivalent to the nonexistence of “eigensolutions.” The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundström [Math. Comp., 26 (1972), pp. 649–686] and others for hyperbolic difference equations.