HODGE DECOMPOSITIONS AND COMPUTATIONAL ELECTROMAGNETICS

The handling of topological aspects in boundary value problems of engineering electromagnetics is often considered to be an engineer's art and not a science. This thesis is an attempt to show that the opposite is true. Through the use of differential forms and rudimentary concepts from homology theory a paradigm variational boundary value problem is formulated and investigated. It is seen that reasoning in terms of the Tonti diagram for this problem may lead to false conclusions if cohomology groups are ignored. As a prelude to this investigation, a suitable orthogonal decomposition of differential forms is derived and the roles played by the long exact homology sequence and topological duality theorems for compact orientable manifolds with boundary are considered in detail.