Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems
Abstract
A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general source 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2023
- DOI:
- arXiv:
- arXiv:2305.06171
- Bibcode:
- 2023arXiv230506171C
- Keywords:
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- Mathematics - Numerical Analysis;
- 65N30;
- 65N12;
- 65N50
- E-Print:
- Accepted for publication in Numerische Mathematik