On the accuracy of finite difference solutions to elliptic partial differential equations

The approximation error of finite-difference methods used to solve elliptic partial differential equations is investigated analytically, examining both the local-interpolation and collocation steps and evaluating the error in approximating the inverse of the differential operator (rather than that in approximating the operator itself). The interpolation operator and the mathematical framework are defined; an upper bound on the approximation error is established; the boundedness of the derivatives is explored; applications to square and equilateral-triangle meshes are shown; and the generalization to the biharmonic equation is considered.