Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms unde...

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Bibliographic Details
Published in:Numerische Mathematik 2023-08, Vol.154 (3-4), p.323-368
Main Authors: Carstensen, Carsten, Nataraj, Neela, Remesan, Gopikrishnan C., Shylaja, Devika
Format: Article
Language:English
Subjects:
Finite element method
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Nonlinearity
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
Vorticity
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Summary:A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D Navier-Stokes equations and the von Kármán equations with Morley, discontinuous Galerkin, C 0 interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-023-01356-w