Construction of Nearly Orthogonal Nedelec Bases for Rapid Convergence with Multilevel Preconditioned Solvers

Abstract

This paper presents a systematic approach to constructing high-order tangential vector basis functions for the multilevel finite element solution of electromagnetic wave problems. The new bases allow easy computation of a preconditioner to eliminate or at least weaken the indefiniteness of the system matrix and thus reduce the condition number of the system matrix. When these bases are used in multilevel solutions, where the multilevels correspond to the order of the basis functions, the resulting p-multilevel-ILU preconditioned conjugate gradient method (MPCG) provides an optimal rate of convergence. We first derive an admissible set of vectors of order p, and decompose this set into two subspaces-rotational and irrotational (gradient). We then reduce the number of vectors by making them orthogonal to all previously constructed lower-order bases. The remaining vectors are made mutually orthogonal in both the vector space and in the range space of the curl operator. The resulting vector basis functions provide maximum orthogonality while maintaining tangential continuity of the field. The zeroth-order space is further decomposed using a scalar-vector formulation to eliminate convergence problems at extremely low frequencies.Numerical experiments show that number of iterations needed for the solution by MPCG is basically constant, regardless of the order of the basis or of the matrix size. Computational speed is improved by several orders of magnitude due to the fast matrix solution of MPCG and to the high accuracy of the higher-order bases.Key words. P -multilevel finite element methods, Maxwell's equations, multilevel preconditioned conjugate gradient method, tangential vector basis functions AMS subject classifications. 65N22, 65N55, 65F10, 83C50 PII. S1064827500367531 1. Introduction. In 1980, Nedelec defined a set of vector finite element basis functions having the properties that they are complete to order p in the range space of the curl operator, that they impose tangential but not normal continuity of the vector, and that they are unisolvent [1].In this paper, we will call finite elements based on these functions Nedelec elements and denote the function space spanned by these elements as E p (curl). Nedelec elements have been shown to be important in the solution of electromagnetic field problems [2], [3]. In particular, [4], [5], [6], [7] showed that these elements eliminate the problem of spurious modes that plague conventional node based approximations of the vector wave equation derived from Maxwell's equations. Numerous authors have derived alternative forms of the Nedelec bases, both for low-order "edge elements" [3], [7], [8], [9], [10], [11], [12] and for high-order tangential vector elements [5], [6], [13], [14], [15], [16], [17], [18], [19], [20], [21]. The goal of these constructions has often been to make the Nedelec bases interpolatory or hierarchical. Hiptmair [22] recently provided a general abstract framework for the systematic construction of these vector bases. With such an abundance of ...

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