Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity

Carstensen, C and Mallik, G and Nataraj, N (2021) Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity. In: IMA Journal of Numerical Analysis, 41 (1). pp. 164-205.

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Abstract

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier-Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton-Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix-Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with L\infty coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM. © 2020 The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Item Type:	Journal Article
Publication:	IMA Journal of Numerical Analysis
Publisher:	Oxford University Press
Additional Information:	The copyright for this article belongs to Oxford University Press
Keywords:	Control nonlinearities; Error analysis; Navier Stokes equations; Viscous flow; Vorticity, 2d navi-stoke equation; A posteriori; Crouzeix-raviart; Elliptic; Morley finite element; Nonconforming; Posteriori; Second orders; Second-order linear nonselfadjoint and indefinite elliptic; Semilinear; Stream function-vorticity formulation; Von Karman equations, Finite element method
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	20 Feb 2023 11:06
Last Modified:	20 Feb 2023 11:06
URI:	https://eprints.iisc.ac.in/id/eprint/80415

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