Integration over Discontinuous Trial Functions in Flux Synthesis

The mathematical difficulties that arise when discontinuous trial functions are substituted into functionals appropriate for continuous functions are investigated by formulating the problem in terms of the mathematically acceptable properties of step functions and their derivatives. The expectable mathematical difficulties are shown to appear in the form of integrals that do not have mathematically defined values. The difficulties can be averted by replacing the integrals with approximate expressions which yield the familiar expressions for coupling across discontinuities. The problem of overdetermining the coupling coefficients appears to be avoidable by consistent application of the approximation.