On the Reduction of Continuous Problems To Discrete Form

A continuous problem, defined as one involving derivatives or integrals, is to be reduced to a discrete problem, involving only algebraic or evaluative operations. An approach involving cells instead of points is taken, and the unknown function is approximated by functional representations, each associated with one cell and an associated set of parameters. Suitable operations are then defined, each associated with a particular cell. These operations remove the configuration coordinates from the problem, leaving only the parameters. Similar operations are defined which link the approximations in adjacent cells, and which translate certain interface conditions to relations between parameters associated with cells. The entire set of relations is then the equivalent of the usual difference equations. A variational algorithm is introduced in order to circumvent certain difficulties associated with matching equations and unknowns. This also permits the convenient retention of certain “exact conditions” associated with the continuous problem. Some illustrative examples are given.