A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes
The discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh-refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it. One approach for this is the careful design of an intermediate tria...
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| Published in | Computers & mathematics with applications (1987) Vol. 68; no. 12; pp. 2167 - 2181 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.12.2014
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0898-1221 1873-7668 |
| DOI | 10.1016/j.camwa.2014.07.019 |
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| Abstract | The discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh-refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it. One approach for this is the careful design of an intermediate triangulation with one-level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation. This paper enfolds the second possibility of some novel discrete Helmholtz decomposition for the nonconforming Morley finite element method. This guarantees the optimality of a standard adaptive mesh-refining algorithm for the biharmonic equation. Numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre-asymptotic range of the adaptive Morley finite element method. |
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| AbstractList | The discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh-refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it. One approach for this is the careful design of an intermediate triangulation with one-level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation. This paper enfolds the second possibility of some novel discrete Helmholtz decomposition for the nonconforming Morley finite element method. This guarantees the optimality of a standard adaptive mesh-refining algorithm for the biharmonic equation. Numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre-asymptotic range of the adaptive Morley finite element method. |
| Author | Gallistl, Dietmar Hu, Jun Carstensen, Carsten |
| Author_xml | – sequence: 1 givenname: Carsten surname: Carstensen fullname: Carstensen, Carsten email: cc@math.hu-berlin.de organization: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany – sequence: 2 givenname: Dietmar surname: Gallistl fullname: Gallistl, Dietmar email: gallistl@math.hu-berlin.de organization: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany – sequence: 3 givenname: Jun surname: Hu fullname: Hu, Jun email: hujun@math.pku.edu.cn organization: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, PR China |
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| Keywords | Kirchhoff–Love plate problem Morley element Discrete Helmholtz decomposition Optimality |
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| SubjectTerms | Algorithms Computer simulation Decomposition Discrete Helmholtz decomposition Finite element method Kirchhoff–Love plate problem Mathematical analysis Mathematical models Morley element Optimality Optimization Triangulation |
| Title | A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes |
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