Error Bounds for Finite Element Methods with Generalized Cubic Splines for a 4-th Order Ordinary Differential Equation with Nonsmooth Data

A boundary value problem for a 4-th order self-adjoint ordinary differential equation is considered in the case where the coefficients of the equation and its right-hand side can be nonsmooth (discontinuous, concentrated or rapidly oscillating functions). Generalized cubic splines of deficiency 1 depending on the major coefficient of the equation are applied. An error analysis of finite element methods exploiting such splines is presented in detail including superconvergence error bounds. This is based on general L–L interpolation error bounds for the splines. 2000 Mathematics Subject Classification: 65D07, 65L10, 65L60, 65L70.