PhD, June 2004
Advisor: Tom Manteuffel
The eddy current model is known as a simplification of Maxwell's equations by neglecting the displacement current. It approximates the Maxwell's equations and explains particular problems encountered in electromagnetism. Under sufficient smoothness, or regularity, assumptions on the eddy current problem, it can be solved accurately with standard conforming finite elements.
However, when these assumptions are violated, the standard finite element method generally fails. It tends to lose global accuracy in the presence of local singularities, for example. In this thesis, we approximately solve the eddy current problem when the coefficients are not smooth and the domain has reentrant edges in three-dimensional space.
To overcome these singularities, we develop a modified first-order system LL* (FOSLL*) method, which is a methodology to solve the first-order partial differential equations using the dual operator L*. Here we show that the modified FOSLL* method allows use of the standard finite element scheme without loss of global accuracy and gives optimal convergence numerically.