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Tangential Magnetic Field Boundary Condition

As boundary conditions, we support the tangential magnetic field given by

\[\left. \mathbf{H} \times \mathbf{n} \right|_{\Gamma_0} = \mathbf{H}_{\text{ext}} \quad,\]

where \(\mathbf{H}_{\text{ext}}\) is the user-defined external magnetic field imposed on the boundary \(\Gamma_0\).

This condition is implemented as a inhomogeneous Neumann condition through a linear form given by

\[f_{\Gamma_0} = \left( \mathbf{n} \times \mathbf{H}_{\text{ext}}, \mathbf{A}' \right)_{\Gamma_0} \quad,\]

where \(\mathbf{A}'\) is a test function in the function space.