A mixed finite element scheme for three-dimensional Maxwell’s transmission eigenvalue problems

Abstract In this paper, we develop an efficient numerical method for solving Maxwell’s transmission eigenvalue problem on three-dimensional domains with vanishing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">b</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> (the first Betti number). To that end, we reduce the order of the fourth-order problem by introducing some auxiliary variables, and ultimately derive an equivalent and stable linear eigenvalue problem with a compact operator. Applying the mixed finite element scheme, we obtain a sparse generalized eigenvalue problem (GEP) without introducing any spurious eigenvalues and achieve the optimal convergence rate for eigenvalues. Moreover, we preprocess the GEP by deflating all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> eigenvalues, while preserving the sparse matrix multiplication structure. With the preprocessing, we develop the block Jacobi–Davidson method with a multigrid-type to calculate a few smallest positive real Maxwell’s transmission eigenvalues. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.