Finite difference approach to fourth-order linear boundary-value problems. (4th September 2020) - Ben-Artzi, Matania; Kramer, Benjamin
Abstract: Discrete approximations to the equation equation*L_contu = u^(4) + D(x) u^(3) + A(x) u^(2) + (A^′(x)+H(x)) u^(1) + B(x) u = f, x[0, 1]equation* 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.