Hierarchical Mixed Finite-Element Method for Maxwell’s Eigenvalue Problems With the Absorbing Boundary Condition and the Surface Current Boundary Condition

A mixed finite-element method (MFEM) is proposed to solve 3-D Maxwell’s eigenvalue problems with the absorbing boundary condition (ABC) and the surface current boundary condition (SCBC). In this MFEM, zero modes (including dc spurious modes and zero physical modes) can be removed using the constrained equations containing the matrices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar {\bar {N}}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tilde {N}$ </tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar {\bar {N}}$ </tex-math></inline-formula> denotes the nullspace matrix for complete and incomplete hierarchical vector basis functions (HVBFs); <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tilde {N}$ </tex-math></inline-formula> is the submatrix of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar {\bar {N}}$ </tex-math></inline-formula>. It can be seen from numerical experiments that the MFEM is more efficient than the tree-cotree technique, the mixed mortar-element method based on a tetrahedral mesh, and the MFEMs based on Gauss’ law for the computational domain with multiple unconnected perfect electric conductor (PEC) boundaries.