Finite difference approach to fourth-order linear boundary-value problems

Discrete approximations to the equation L_contu = u⁽⁴⁾ + D(x)u⁽³⁾ + A(x)u⁽²⁾ + (Á(x) + H(x))u⁽¹⁾ + B(x)u = f, x ∈ [0, 1] are considered. This is an extension of the Sturm–Liouville case D(x) ≡ H(x) ≡ 0 (Ben-Artzi et al. (2018) Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal., 38, 1485–1522) to the non-self-adjoint setting. The ‘natural’ boundary conditions in the Sturm–Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second-, third- and fourth-order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties of compactness and coercivity. It allows us to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.