Lowest order Virtual Element approximation of magnetostatic problems
For computational electromagnetics, this extends lowest order edge finite elements to general polyhedral meshes, offering a robust and accurate method.
The paper presents a simplified lowest order Serendipity Virtual Element method for solving linear magnetostatic problems in 3D, using edge-based magnetic field and vertex-based Lagrange multiplier unknowns. Numerical examples show good accuracy and robustness on polyhedral meshes.
We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=μ\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind Nédélec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions.