A Tearing and Interconnecting Formulation for Magneto-Quasi-Statics

TL;DR

This paper introduces a novel tearing and interconnecting domain decomposition method for magneto-quasi-static problems, effectively handling interface coupling and kernel components in conducting and insulating regions.

Contribution

It proposes a space splitting approach that ensures invertibility of subdomain operators and maintains a globally H(div) magnetic field despite discontinuities.

Findings

The formulation eliminates kernel components in the non-conducting domain.

Both subdomain operators are proven to be invertible.

The magnetic field remains globally in H(div) despite potential discontinuities.

Abstract

This note deals with a tearing and interconnecting (special non-overlapping domain decomposition) formulation for magneto-quasi-statics (also known as the eddy current model). Only two subdomains are considered, one conducting and one insulating. Using a straightforward tree-cotree splitting, one can get rid of some kernel components in the non-conducting region, but due to the coupling across the interface, a lot of kernel functions remain that are associated with the interface. The formulation presented here overcomes this problem by using a space splitting into gradient fields and a complementary space. Under a mild condition on that splitting, it is shown that (i) one does not need any gradient part in the non-conducting domain, and therefore no coupling of any gradient components between the two subdomains, (ii) both subdomain operators are invertible, and (iii) although the magnetic vector potential is discontinuous across the subdomain interface, the corresponding magnetic field is globally in H(div).