CONVERGENCE OF THE HIPTMAIR-XU PRECONDITIONER FOR MAXWELL'S EQUATIONS WITH JUMP COEFFICIENTS (I): EXTENSIONS OF THE REGULAR DECOMPOSITION

This paper is the first in a series of two articles, aiming to prove the convergence of the HX preconditioner originally proposed by Hiptmair and Xu [SIAM T. Numer. Anal., 45 (2007), pp. 2483-2509] for Maxwell's equations with jump coefficients. In this paper, we establish several extensions of the discrete regular decomposition for edge finite element functions defined in three-dimensional domains. The functions defined by the new discrete regular decompositions can inherit zero degrees of freedom of the considered edge finite element function on some faces and edges of polyhedral domains as well as of some non-Lipschitz domains and possess nearly optimal stability with only a logarithmic factor.

Automated summary

This paper proves the convergence of the HX preconditioner proposed by Hiptmair and Xu for Maxwell's equations, and establishes extensions of the discrete regular decomposition for edge finite element functions in three-dimensional domains. The new functions defined by the discrete regular decompositions inherit zero degrees of freedom of the edge finite element function in polyhedral domains and possess nearly optimal stability with only a logarithmic factor.