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High order Nédélec elements with local complete sequence properties. (English) Zbl 1135.78337

Summary: The goal of the present work is the efficient computation of Maxwell boundary and eigenvalue problems using high order \(H\)(curl) finite elements.
The authors discuss a systematic strategy for the realization of arbitrary order hierarchic \(H\)(curl)-conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowest order Nédélec, higher-order edge-based, face-based (only in 3D) and element-based ones.
Our new shape functions provide not only the global complete sequence property but also local complete sequence properties for each edge-, face-, and element-block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block-diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradient basis functions can be skipped, which reduces the problem size, and improves the condition number.

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
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References:

[1] DOI: 10.1002/nme.847 · Zbl 1042.65088
[2] DOI: 10.1007/PL00005386
[3] DOI: 10.1007/s002110000182 · Zbl 0967.65106
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[8] DOI: 10.1109/8.791939 · Zbl 0955.78014
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