Preconditioners for the indefinite linear system arising from the hp discretization of Maxwell's equations

Abstract

SUMMARYThis paper describes a preconditioner for the iterative solution to the indefinite linear equation system that is obtained when the vector wave equation is discretized by higher-order H(curl)-conforming finite elements. The preconditioner is based on a decomposition of the H(curl)-conforming functions into blocks comprising lower-order functions and higher-order functions. Fast convergence is obtained when the grid spacing is sufficiently small. A method for selecting suitable grid spacings is proposed and a series of numerical examples are included to demonstrate the effectiveness of the approach.

An Investigation of a Preconditioner for High-Frequency Maxwell Problems

“…In the above, B D A C jÄjM and Q B follow from the static condensation of the interior block of B in the fictional linear system By D 0. The reason for following this approach is that the resulting blocks Q B EE and Q B FF are then positive definite (Ledger, 2009). If an additional decomposition of the face functions into gradients and nongradients is undertaken, the block associated with the gradient face functions can then be treated in similar way as the high order edge block in Eq.…”

“…In Ledger (2009), numerical results were presented that showed that the eigenvalues of the preconditioned system became clustered about a point bounded away from zero once the mesh spacing was sufficiently small. In this section, a detailed investigation of the eigenspectrum of the preconditioners proposed in Eqs.…”

“…Based on previous experience (Ledger, 2009), the use of a modified version of Zaglmayr's preconditioner (Zaglmayr, 2006) is proposed for the iterative solution of Kx D r . The preconditioner takes the form C 1 with…”

“…The work extends previous work by Zaglmayr (2006), which proposed a block diagonal preconditioner for the positive definite system matrix that is obtained in eddy current problems and regularized magnetostatic problems. It also builds on a previous study by Ledger (2009), which presented some initial two-dimensional numerical experiments for an indefinite preconditioner for Eq. (3).…”

hp-Finite element simulation of three-dimensional eddy current problems on multiply connected domains

“…The hierarchic basis functions proposed by Schöberl and Zaglmayr lend themselves to a parameter-robust iterative two-level solver: Assuming that an appropriate h-version solver on the low-order space (the coarse level) is available, simple block Jacobi smoothing is sufficient on the high-order level (the fine level). This preconditioning technique is described and analysed in [65,78] and has been modified for wave propagation problems by Ledger [48,47]. In this work, we combine and extend these strategies by further exploiting the discrete space splitting of the underlying hierarchical hp-finite element basis and suggest an improved version of a hierarchical hp-finite element solver for eddy current problems.…”