Reduced first order differential equation with optimal control finite element penalty functions

The basic idea of optimal control penalty function method is applied to reduced first order differential equation. In contrast to a popular practice of the penalty function method being utilized as a scheme to impose an incompressibility condition in solid or fluid mechanics, the present study demonstrates that the concept of penalty functions can be generalized so as to overcome some of the computational difficulties such as occur in non-self-adjoint differential equations. Specific applications are made to convection-radiation heat transfer and boundary layer problems. The encouraging results reported herein are expected to promote the need for a full scale investigation of mathematical error estimates in the future.