Timon Seibel

Timon Seibel, Sebastian Schöps, Kersten Schmidt

In this work, we study the application of a hybrid mixed domain decomposition (HMDD) method for the rotor-stator coupling of a permanent magnet synchronous machine. For this, we derive a variational formulation on the electric machine inspired by hybridized discontinuous Galerkin methods using a mixed magnetostatics problem, an affine material law and boundary conditions respecting the symmetry of the motor. We are then able to locate the resulting finite element method within the HMDD framework presented in arXiv:2604.22543. This enables us naturally to transfer the well-posedness results and error estimates for the HMDD method to the finite element method considered in this work. Lastly, as a proof of concept, we consider an academic example and compare the resulting magnetic flux density and potential lines to their counterparts obtained by a well-established in-house code using iso-geometric analysis.

Kersten Schmidt, Timon Seibel, Sebastian Schöps

We present a domain decomposition formulation based on hybridization which is inspired by hybridized discontinuous Galerkin (HDG) methods, that enhance mixed domain decomposition methods by incorporating stabilization terms. Unlike discontinuous Galerkin methods, our analysis of the proposed finite element method is based on a corresponding consistent variational formulation and a perturbed Galerkin method. In the variational formulation the divergence appears not only within subdomains, but also as an $L^2$-surface quantity on the interfaces. Furthermore, the traces of the finite element functions on the interfaces are replaced by $L^2$-distributions. The well-posedness of the perturbed Galerkin method is shown for an appropriate choice of subspaces, in a manner similar to that of the variational formulation. For the finite element method we use Raviart-Thomas elements for the dual variable and piecewise polynomials for the primal and hybrid variables, respectively. We perform an analysis of the discretization error which is explicit in the stabilization parameter $τ$. Numerical experiments for piecewise smooth solutions using finite element spaces of order~$q$ on curved quadrilateral meshes confirm the predicted convergence rate of $q+1$ for small values of $τ$. In the error analysis we observe the discretization error to be uniformly bounded in $τ$. Even for large $τ$ values the observed convergence rates for the primal and for the hybrid variables are $q+1$. For the dual variable the convergence rate depends on the stabilization parameter and the mesh-width, with an asymptotic rate of $q+\tfrac12$.