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

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.