Adaptive Computation of fourth order problems
A talk in the BI.discrete Workshop series by
Carsten Carstensen from Berlin
| Abstract: | The popular (piecewise) quadratic schemes for the fourth-order plate bending problems
based on triangles are the nonconforming Morley finite element, two discontinuous Galerkin,
the C0 interior penalty, and the WOPSIP schemes. The first part of the presentation discusses
recent applications to the linear bi-Laplacian and to semi-linear fourth-order problems like
the stream function vorticity formulation of incompressible 2D Navier-Stokes problem and the von Kármán plate bending problem. The role of a smoother is emphasised and reliable and efficient
a posteriori error estimators give rise to adaptive mesh-refining strategies that recover optimal
rates in numerical experiments.
The second part concerns conforming finite elements for the fourth-order plate problem:
The hierarchical Argyris finite element method allows for rate optimal adaptive multigrid
algorithms.
The presentation concerns ongoing joint work with Neela Nataraj of IITB in Mumbai and
Benedickt Gräßle of HU in Berlin. Related references include the following. C. Carstensen, B. Gräßle, and N. Nataraj: Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation, Journal of Numerical Mathematics, electronic, 2023. arXiv:2310.05648 C. Carstensen, B. Gräßle, and N. Nataraj: A posteriori error control for fourth-order semilinear problems with quadratic nonlinearity, submitted to SINUM, arXiv:2309.08427 C. Carstensen, Carsten and Jun Hu: Hierarchical Argyris finite element method for adaptive and multigrid algorithms, Comput. Methods Appl. Math., volume 21, pp. 529–556, 2021. C. Carstensen, N. Nataraj: A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides, Comput. Methods Appl. Math., volume 21, pp. 289–315, 2021. C. Carstensen, N. Nataraj, G.C. Remesan, D. Shylaja Lowest-order FEM for fourth-order semi-linear problems with trilinear nonlinearity, Numerische Mathematik 154, pages 323–368, 2023. C. Carstensen, N. Nataraj Lowest-order equivalent nonstandard finite ele- ment methods for biharmonic plates , ESAIM: Mathematical Modelling and Numerical Analysis, 56(1), 41–78 (2022) B. Gräßle: Optimal multilevel adaptive FEM for the Argyris element, Computer Methods in Applied Mechanics and Engineering, volume 399, pp. 115352, 2022. Within the CRC this talk is associated to the project(s): A7, B3, B7 |