Chapter

Nonconforming finite element methods

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations

Published in print October 2010 | ISBN: 9780199577040
Published online January 2011 | e-ISBN: 9780191595172 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199577040.003.0005

Series: Numerical Mathematics and Scientific Computation

Nonconforming finite element methods

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Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C1 FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ2 for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.

Keywords: nonconforming FEMs; fully nonlinear equations; variational methods for eigenvalue problems; convergence theory; monotone problems; quasilinear problems; nonlinear elliptic problems; nonlinear boundary conditions; quadrature approximate FEMs

Chapter.  33702 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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