×

Dual-grid-based tree/cotree decomposition of higher-order interpolatory \(H(\nabla \wedge, \Omega )\) basis. (English) Zbl 1193.65197

Summary: This work extends the zeroth-order tree/cotree (TC) decomposition method into higher order (HO) interpolatory elements and develops the constraints operator required for the elimination of spurious solutions for general HO spectral basis. Earlier methods explicitly enforce the divergence condition that requires a mixed finite element (FE) formulation with both \(H^{1}\) and \(H(\nabla \wedge)\) expansions and involves repeated solutions of the Poisson equation. A recent approach, which avoids the mixed formulation and the Poisson problem, uses TC decomposition of edge DoF over the primal graph and construction of integration and gradient matrices. The approach is easily applied to HO hierarchical elements but becomes quite complex for HO spectral elements. In the presence of internal DoF, it is difficult to utilize the primal graph for an explicit decomposition of the spectral DoF. In contrast, this work utilizes the dual grid, resulting in an explicit decomposition of DoF and construction of constraint equations from a fixed element matrix. Thus, mixed formulation and the Poisson problems are avoided while eliminating the need for evaluation of integration and gradient matrices. The proposed constraints matrix is element-geometry independent and possesses an explicit sparsity formulation reducing the need for dynamic memory allocation. Numerical examples are included for verification.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems

Software:

Trilinos
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nedelec, Mixed finite elements in \(\mathbb{R}\)3, Numerische Mathematik 35 pp 315– (1980) · Zbl 0419.65069
[2] Nedelec, A new family of mixed finite elements in \(\mathbb{R}\)3, Numerische Mathematik 50 pp 57– (1986)
[3] Manges, Tree-cotree decomposition for first-order complete tangential vector basis, International Journal for Numerical Methods in Engineering 40 pp 1667– (1997)
[4] Venkatarayalu, Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators, IEEE Transactions on Microwave Theory and Techniques 54 (7) pp 3019– (2006)
[5] Lee, Hierarchical vector finite elements for analyzing waveguiding structures, IEEE Transactions on Microwave Theory and Techniques 51 (8) pp 1897– (2003)
[6] Perepelitsa, Finite element analysis of arbitrary shaped cavity resonator using H1(curl) elements, IEEE Transactions on Magnetics 33 (3) pp 1776– (2004)
[7] White DA, Koning JM. Computing solenoidal eignemodes of the vector Helmholtz equation: a novel approach. IEEE International AP-S Symposium Dig., Washington, DC, July 2005; 193-196.
[8] Golan, Foundations of Linear Algebra (1995) · doi:10.1007/978-94-015-8502-6
[9] Oden, Applied Functional Analysis (1996) · Zbl 0851.46001
[10] Monk, Computational Electromagnetics pp 127– (2003) · doi:10.1007/978-3-642-55745-3_9
[11] Girault, Finite Element Approximation of the Navier-Stokes Equations (1979) · Zbl 0413.65081
[12] Wilton, Higher order interpolatory vector basis for computational electromagnetics, IEEE Transactions on Antennas and Propagat 45 (3) pp 329– (1997)
[13] Sadiku, Numerical Techniques in Electromagnetics (2000) · doi:10.1201/9781420058277
[14] Solin, Higher Order Finite Elements (2004)
[15] Dufresne M, Silvester PP. Universal matrices for the N-dimensional finite element. Third International Conference on Computation in Electromagnetics, Bath, England, 10-12 April 1996; 223-228.
[16] Villeneuve, Universal matrices for high order finite elements in nonlinear magnetic field problems, IEEE Transactions on Magnetics 33 (5) pp 4131– (1997)
[17] Heroux M, Bartlett R, Howle V, Hoekstra R, Hu J, Kolda T, Lehoucq R, Long K, Pawlowski R, Phipps E, Salinger A, Thornquist H, Tuminaro R, Willenbring J, Williams A. An Overview of Trilinos. Technical Report, SAND2003-2927, Sandia National Laboratories, 2003. · Zbl 1136.65354
[18] Balanis, Advanced Engineering Electromagnetics (1989)
[19] Taflove, The Finite-Difference Time Domain Method in Computational Electrodynamics (1999) · Zbl 0840.65126
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.