Second-order derivatives of the general-purpose finite element package SEPRAN via source transformation

Second-order derivatives are crucial ingredients to a variety of numerical methods. Often, they are difficult to get with numerical differentiation by divided differencing. Automatic differentiation provides an alternative by a program transformation capable of evaluating Jacobians, Hessians, or higher-order derivatives of functions given in the form of computer programs. SEPRAN is a general-purpose finite element package written in Fortran 77 used in various scientific areas ranging from fluid dynamics to structural mechanics to electromagnetism. By transforming SEPRAN twice using the automatic differentiation tool ADIFOR, second-order derivatives are evaluated without truncation error. Numerical experiments are reported in which second-order derivatives of a flow field with respect to an inflow velocity are computed, demonstrating the feasibility of this approach.