Fast algorithms of edge element discretizations and adaptive finite element methods for two classes of Maxwell equations

Abstract

Fast algorithms and adaptive methods for edge element discretizations of Maxwell equations are two very active areas today in computational electromagnetism. Fast algorithms for edge element discretizations and adaptive finite element methods for two classes of typical Maxwell equations are developed. Firstly, the fast iterative methods and efficient preconditions for high order edge discretizations of <italic>H</italic>(curl)-elliptic equations are constructed by using the stable decompositions of high order edge finite element spaces, we prove that both the convergent rate of iterative methods and the condition numbers of our preconditioners are independent of mesh size. Secondly, quasi-optimal error estimates in both <italic>L</italic>²-norm and <italic>H</italic>(curl)-norm for edge discretizations of the time-harmonic Maxwells equations are obtained by using discrete Helmholtz decompositions, the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function, we also construct and analysis the corresponding two-grid method. Finally, we consider the standard Adaptive Edge Finite Element Method (AEFEM) for the <italic>H</italic>(curl)-elliptic equations with variable coefficients and the indefinite time-harmonic Maxwells equations, respectively. As is customary in practice, AEFEM marks exclusively according to the error estimator without special treatment of oscillation and performs a minimal element refinement without the interior node property. We prove that the AEFEM is convergent. Then using this geometric decay, global lower bounds and localized upper bound of a residualtype error estimate, we derive the quasi-optimal cardinality of the AEFEM.