Discrete fourth-order Sturm-Liouville problems

A discrete fourth-order elliptic theory on a one-dimensional interval is constructed. It is based on 'Hermitian derivatives' and compact higher-order finite difference operators, and is shown to possess the analogues of the standard elliptic theory such as coercivity and compactness. The discrete version of the fourthorder Sturm-Liouville problem (d/dx)⁴ u + d/dx(A(x)d/dxu) + B(x)u = f on a real interval is studied in terms of the functional calculus. The resulting (compact) finite difference scheme constitutes a scale of finitedimensional Sturm-Liouville problems. A major difficulty is the presence of boundaries, in contrast to periodic problems (and analogous to boundary layers in Navier-Stokes simulations). Convergence of the finite-dimensional solutions to the continuous one is proved in the general case, and optimal (O(h⁴)) convergence rates are obtained in the constant coefficient case. Numerical examples are given, demonstrating the optimal rate even in highly oscillatory cases.