Auto-Gauging of Vector Potential by Parallel Sparse Direct Solvers—Numerical Observations

When using magnetic vector potential (MVP)-based formulations for magnetostatic or eddy-current problems, either gauge conditions specifying the divergence of the MVP or tree gauging by eliminating redundant degrees of freedom of the MVP is usually imposed to ensure uniqueness of solutions. Explicit gauging of the MVP is not always necessary since classical iterative solvers can automatically and implicitly fix the gauge as long as the right-hand side vectors are consistent. Besides iterative solvers, implicit gauging is also observed when using state-of-the-art parallel sparse direct solvers (PSDSs), thanks to the built-in functions of handling null-spaces of either real symmetric positive semi-definite matrix systems or those complex symmetric systems from eddy-current problems. Both static and eddy-current examples are solved by PSDS to demonstrate results of local physical quantities or global quantities such as magnetic energy or joule losses. High-order edge/nodal elements are also considered in our numerical examples and it is observed that PSDS can also easily and correctly handle the delicate discrete null spaces.