Carsten Carstensen: Monday, 10th September at 11:00
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 last part addresses recent developments on adaptive multilevel Argyris finite element methods. The presentation is based on joint work with B. Gräßle, (Humboldt) and N. Nataraj (IITB, Mumbai) partly reflected in the references below.
REFERENCES
[1] C. Carstensen, B. Gräßle,, and N. Nataraj. Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation, J. Numer. Math., volume 32, pp. 77–109, 2024, arXiv:2310.05648.
[2] C. Carstensen, B. Gräßle,, and N. Nataraj. A posteriori error control for fourth-order semilinear problems with quadratic nonlinearity, SIAM J. Numer. Anal., volume 62, pp. 919–945, 2024.
[3] C. Carstensen, Jun Hu. Hierarchical Argyris finite element method for adaptive and multigrid algorithms, Comput. Methods Appl. Math., volume 21, pp. 529–556, 2021.
[4] 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.
[5] C. Carstensen, N. Nataraj, G.C. Remesan, D. Shylaja. Lowest-order FEM for fourth-order semi-linear problems with trilinear nonlinearity, Numerische Mathematik 154, pp. 323–368, 2023.
[6] C. Carstensen, N. Nataraj. Lowest-order equivalent nonstandard finite element methods for biharmonic plates, ESAIM: Mathematical Modelling and Numerical Analysis, 56(1), 41–78, 2022.
[7] B. Gräßle,. Optimal multilevel adaptive FEM for the Argyris element, Computer Methods in Applied Mechanics and Engineering, volume 399, pp. 115352, 2022.
