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FROM:
Alexander Düster, Ernst Rank and Barna Szabó
(1) Hamburg University of Technology, Hamburg, Germany
(2) Institute for Advanced Study, Technische Universitaet Muenchen, Muenchen, Germany
(3) Washington University, St. Louis, MO, USA
"The p-Version of the Finite Element and Finite Cell Methods". In: Encyclopedia of Computational Mechanics, 2nd edition, Vol. 2: Solids and Structures, Chapter 4. Editors: Erwin Stein, Rene de Borst and Thomas J.R. Hughes, John Wiley & Sons Ltd, 2017.
ABSTRACT: In the first part of this chapter the basic algorithmic structure and performance characteristics of the p-version of the finite element method are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p-version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown that p-extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any quantity of interest. In the second part the p-version of the finite element method will be extended to a high-order fictitious domain approach, the finite cell method. Whereas the finite cell method inherits the advantages of the p-version with respect to accuracy and robustness, it relieves analysts from the necessity to generate finite element meshes. Thus, it strongly supports the analysis of problems with highly complicated geometry, for which meshing with finite elements would be very difficult. Given the growing demand for verified numerical solutions, the p- version of the finite element and finite cell methods are expected to play an increasingly important role.
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