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On the \(p\)- and \(hp\)-extension of Nédélec’s curl-conforming elements. (English) Zbl 0820.65066
The author establishes error estimates in terms of both mesh size \((h)\) and element interpolation order \((p)\) for J. C. Nédélec’s “edge” elements [Numer. Math. 39, 97-112 (1982; Zbl 0488.76038)] applied to the vector potential formulation of a magnetostatics problem. The author’s focus is on curl-conforming elements, but he also provides estimates for divergence-conforming elements during the course of the proofs.
The author achieves his goal of demonstrating the Babuška-Brezzi conditions for edge elements of varying order and size. He does this with the assumptions that the elements are hexahedra which are affine images of a unit cube and that the solution region is a convex polyhedron. The line of reasoning is to first employ Legendre polynomial expansions to establish variable order estimates on the unit cube and to extend these estimates to elements of variable size by scaling arguments. The convexity requirement arises in constructing an auxiliary divergence-free vector function in \((H^ 1)^ 3\) with specified curl.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
78A30 Electro- and magnetostatics
35Q60 PDEs in connection with optics and electromagnetic theory
Software:
TRIFOU
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References:
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