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Finite Element-Based Algorithms To Make Cuts for Magnetic Scalar Potentials: Topological Constraints and Computational Complexity - Abstract

This paper outlines a generic algorithm to generate cuts for magnetic scalar potentials in 3-dimensional multiply-connected finite element meshes. The algorithm is based on the algebraic structures of (co)homology theory with differential forms and developed in the context of the finite element method and finite element data structures. The paper also studies the computational complexity of the algorithm and examines how the topology of the region can create an obstruction to finding cuts in O(m2 0) time and O(m0) storage, where m0 is the number of vertices in the finite element mesh. We argue that in a problem where there is no a priori data about the topology, the algorithm complexity is O(m2 0) in time and O(m4/3 0) in storage. We indicate how this complexity can be achieved in implementation and optimized in the context of adaptive mesh refinement.