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The fast solution of three-dimensional eddy current problems is still an open problem, especially when real-size finite element models with millions of degrees of freedom are considered. In order to lower the number of degrees of freedom a magnetic scalar potential can be used in the insulating parts of the model. This may become difficult when the model geometry presents some conductive parts which are multiply connected. In this work a multigrid-based algoritm is proposed that allows for a calculation in linear-time of cohomology, which is needed to introduce the scalar potential without cuts. This algorithm relies on an algebraic multigrid solver for curl-curl field problems, which ensures optimal computational complexity. Numerical results show that the novel algorithm outperforms state-of-the-art methods for cohomology generation based on homological algebra. In addition, based on this algoritm, novel <inline-formula> <tex-math notation="LaTeX">$a,v$ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\varphi $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\varphi $ </tex-math></inline-formula> formulations to analyze three-dimensional eddy current problems in multiply connected domains are proposed. Both formulations, after discretization by the cell method, lead to a complex symmetric system of linear equations amenable to fast iterative solution by Krylov-subspace solvers. These formulations are able to provide very accurate numerical results, with a minimum amount of degrees of freedoms to represent the eddy current model. In this way the computational performance is improved compared to the classical <inline-formula> <tex-math notation="LaTeX">$A,V$ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> formulation typically implemented in finite element software for electromagnetic design.

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Fast Solution of 3-D Eddy-Current Problems in Multiply Connected Domains by a, v-&#x03C6; and t-&#x03C6; Formulations With Multigrid-Based Algorithm for Cohomology Generation