%
% Fill only lines marked with dots  
% 
%
%
\documentclass[10pt,a4paper,twoside]{article}
\usepackage{latexsym,amssymb}
\usepackage{amssymb,amsmath}
\usepackage{graphicx}
\usepackage{color}
\usepackage{fancyhdr}
\pagestyle{fancy}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  PLEASE, ONLY IN ENGLISH                                         %
%  PLEASE, ONLY ONE PAGE                                           %
%  PLEASE, USE LATEX2E                                             %
%                                                                  %
%  PLEASE, DO NOT DEFINE NEW COMMANDS                              %
%  USE NEWTHEOREMS BELOW, PLEASE, IF YOU WRITE THEOREMS AND SUCH   % %                                                                  %
%  PLEASE, CONTROL THE SIZE OF YOUR ABSTRACT BEFORE YOU SEND IT    %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newtheorem{thm}{Theorem}
\newtheorem{defi}[thm]{Definition}
\newtheorem{axi}[thm]{Axiom}
\newtheorem{clm}[thm]{Claim}
\newtheorem{col}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{con}[thm]{Conjecture}
\newtheorem{hyp}[thm]{Hypothesis}
\newtheorem{rem}[thm]{Remark}
\newcommand{\proof}{\noindent{\bf Proof:}\hspace{0.2cm}}
\setcounter{thm}{0}
\newcommand{\N}{{\mathbb N}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\W}{W_0^{1,p}(\Omega)}
\newcommand{\eps}{{\varepsilon}}
\newcommand{\ds}{\displaystyle}

\newcommand{\F}{\mathcal F}
\newcommand{\E}{\mathscr E}
\newcommand{\J}{\mathscr{J}}
\newcommand{\Pl}{(\mathscr{P})}
\newcommand{\CC}{\mathbb C}
\newcommand{\into}{\int\limits_{\Omega}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textwidth 15.0cm
\textheight 22.0cm
\oddsidemargin 0.4cm
\evensidemargin 0.4cm
\topmargin  0.0cm

\frenchspacing

\headwidth=\textwidth 	
\fancyhead[HCO]{8th Int. Conf. on Mathematics and Informatics,
	September 9-10, 2021, T\^argu Mure\c s, Romania}
\lhead{}\rhead{}
\fancyfoot{}
%


\begin{document}

\begin{center}{\Large\bf
Operator-based iterative methods for nonlinear elliptic PDEs}
\end{center}
%
\begin{center}{\large\bf\noindent
J\'anos Kar\'atson}
\\[2mm]
%%
{ELTE University/NumNet Research Group \& Technical University,   Budapest, Hungary}
%%
\\[1mm]\texttt{
%%	
kajkaat@caesar.elte.hu
%%
}\end{center}
%

\vspace*{7mm}
Nonlinear elliptic problems arise in various physical and other applications where the model describes a stationary state of the given process, such as flow problems, elasticity or glaciology etc. 
The widespread way to solve such problems is a finite element discretization  plus some Newton-like iteration. The talk summarizes earlier and recent work on an operator approach, where quasi-Newton methods are constructed via spectral equivalence, also interpreted as variable preconditioning.  The considered problem classes depend on the structural conditions of the nonlinearity. The robustness of the method is illustrated by numerical tests for some real-life models.
 

\begin{thebibliography}{9} 
 \bibitem{1.} 
 I. Farag\'o,  J. Kar\'atson,  \textit{Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Applications}, NOVA Science Publishers, 2002.
\bibitem{2.} 
B. Borsos, J. Kar\'atson, Quasi-Newton variable preconditioning for nonlinear nonuniformly monotone elliptic problems posed in Banach spaces,
\textit{IMA J. Numer. Anal.} (2021), \texttt{https://doi.org/10.1093/imanum/drab024}
\end{thebibliography}
%

\end{document}


