zbMATH — the first resource for mathematics

Fully automatic \(hp\)-adaptivity for Maxwell’s equations. (English) Zbl 1063.78019
Summary: I report on the development of a fully automatic \(hp\)-adaptive strategy for the solution of time-harmonic Maxwell equations. The strategy produces a sequence of grids that deliver exponential convergence for both regular and singular solutions. Given a (coarse) mesh, we refine it first globally in both \(h\) and \(p\), and solve the problem on the resulting fine mesh. We consider then the projection-based interpolants of the fine mesh solution with respect to both current and next (to be determined) coarse grid, and introduce the interpolation error decrease rate equal to the difference of the old and new (coarse) mesh interpolation errors vs. number of degrees-of-freedom added. The optimal \(hp\)-refinements leading to the next coarse grid are then determined by maximizing the interpolation error decrease rate.

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A25 Electromagnetic theory (general)
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
[1] Babuška, I.; Craig, A.; Mandel, J.; Pitkaränta, J., Efficient preconditioning for the p-version finite element method in two dimensions, SIAM J. numer. anal., 28, 3, 624-661, (1991) · Zbl 0754.65083
[2] Beck, R.; Deuflhard, P.; Hiptmair, R.; Hoppe, R.H.W.; Wohlmuth, B., Adaptive multilevel methods for edge element discretizations of maxwell’s equations, Surv. meth. ind., 8, 271-312, (1999) · Zbl 0939.65136
[3] Demkowicz, L.; Monk, P.; Vardapetyan, L.; Rachowicz, W., De Rham diagram for hp finite element spaces, Math. comput. appl., 39, 7-8, 29-38, (2000) · Zbl 0955.65084
[4] Demkowicz, L.; Babuška, I., Optimal p interpolation error estimates for edge finite elements of variable order in 2D, SIAM J. numer. anal., 41, 4, 1195-1208, (2003) · Zbl 1067.78016
[5] Demkowicz, L.; Rachowicz, W.; Devloo, Ph., A fully automatic hp-adaptivity, J. sci. comput., 17, 1-3, 127-155, (2002) · Zbl 0999.65121
[6] L. Demkowicz, 2D hp-Adaptive Finite Element Package (2Dhp90). Version 2.1, TICAM Report 02-06
[7] Demkowicz, L., hp-adaptive finite elements for time-harmonic Maxwell equations, () · Zbl 1059.78032
[8] D. Pardo, L. Demkowicz, Integration of hp-adaptivity and multigrid. I. A two grid solver for hp finite elements, TICAM Report 02-33
[9] W. Rachowicz, L. Demkowicz, L. Vardapetyan, hp-Adaptive FE modeling for Maxwell’s equations, Evidence of Exponential Convergence ACES’ 99, Monterey, CA, 16-20 March 1999
[10] A. Zdunek, W. Rachowicz, A three-dimensional hp-adaptive finite element approach to radar scattering problems, in: Fifth World Congress on Computational Mechanics Vienna, Austria, 7-12 July 2002 · Zbl 1063.78024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.