Hierarchical Universal Matrices for Curvilinear Tetrahedral H(curl) Finite Elements With Inhomogeneous Material Properties

A general method for calculating mass and stiffness matrices of
$ H(\mathop {\mathrm {curl}})$
-conforming finite elements (FEs) is proposed. It applies to curvilinear geometries and/or inhomogeneous materials, preserves the nullspace of the curl operator, and is computationally efficient. In the FE integrals, the terms incorporating the effects of geometry and materials are expanded in a series of multivariate polynomials. As a result, the element matrices can always be integrated analytically by means of predefined universal matrices. The mapping from the reference element to the physical configuration is done by suitable forms of the Piola transformation so that the resulting FE bases are guaranteed to preserve the nullspace of the curl operator. A mathematical proof, including the curvilinear case, is included. The suggested approach features a representation limit: When the metric expansion is continued beyond a critical order, which is determined by the order of the FE basis, the truncation error becomes zero. Hence, the number of universal matrices is bounded, and the sole source of error is the numerical calculation of the expansion coefficients for the metric terms. The method is validated by numerical examples, and its computational efficiency is demonstrated by a comparison to competing approaches.