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Finite Element Methods for Maxwell's Equations$
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Peter Monk

Print publication date: 2003

Print ISBN-13: 9780198508885

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198508885.001.0001

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Finite Elements on Tetrahedra

Finite Elements on Tetrahedra

Chapter:
(p.99) 5 Finite Elements on Tetrahedra
Source:
Finite Element Methods for Maxwell's Equations
Author(s):

Peter Monk

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198508885.003.0005

The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in practice. Details of the constructions of scalar and vector finite elements of all orders are presented. The vector elements are due to Nedelec. In particular, the curl-conforming elements of this chapter are the widely used ‘edge-elements’ whereas the corresponding divergence-conforming elements are often termed ‘face elements’ (they are extensions to 3D of the Raviart-Thomas elements). The appropriate conforming and unisolvence properties of the elements are proven, and the important discrete de Rham diagram relating the interpolation operators for these finite elements with the divergence, gradient, and curl operators are verified; this is used heavily in later theory. Interpolation error estimates under mesh refinement are derived (h-version of the finite element method). A convenient basis for linear and quadratic finite elements is presented, and spaces of elements on boundaries of the domain are briefly discussed.

Keywords:   edge elements, face elements, interpolation error estimates, de Rham diagram, unisolvence

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