Second‐order ordinary differential equations
This chapter develops finite-element models for a class of boundary-value problems governed by a second-order ordinary differential equation. It considers a standard form of the differential equation which is known as Model Problem. Formulation of the finite element method using global basis functions establishes a strong connection between the weighted residual method or the weak form and the finite element method. However, construction of global basis functions and evaluations of integrals are not practical for complex geometries particularly for multidimensional problems. Boundary conditions are specified on the boundary of the whole domain, and no boundary conditions are available for any of the elements except for an element whose boundary falls on the global boundary. The weak form before application of boundary conditions Equation is valid for each element provided that the integrals are evaluated over the element domain only and the secondary variables are evaluated at the element boundary points.