scholarly journals GRADIENT SCHEMES: A GENERIC FRAMEWORK FOR THE DISCRETISATION OF LINEAR, NONLINEAR AND NONLOCAL ELLIPTIC AND PARABOLIC EQUATIONS

2013 ◽  
Vol 23 (13) ◽  
pp. 2395-2432 ◽  
Author(s):  
JEROME DRONIOU ◽  
ROBERT EYMARD ◽  
THIERRY GALLOUET ◽  
RAPHAELE HERBIN
Keyword(s):  
Parabolic Equations ◽  
Degrees Of Freedom ◽  
Parabolic Problems ◽  
Finite Difference Schemes ◽  
Nonlocal Operators ◽  
Nonlinear Elliptic ◽  
Generic Framework ◽  
Mixed Family ◽  
Elliptic And Parabolic Equations ◽  
Nonconforming Methods

Gradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformity and compactness, are shown in this paper to be sufficient to prove the convergence of gradient schemes for linear and nonlinear elliptic and parabolic problems, including the case of nonlocal operators arising for example in image processing. We also show that the schemes of the Hybrid Mimetic Mixed family, which include in particular the Mimetic Finite Difference schemes, may be seen as gradient schemes meeting these four properties, and therefore converges for the class of above-mentioned problems.

scholarly journals Block monotone iterations for solving coupled systems of nonlinear parabolic equations

2020 ◽  
Vol 61 ◽  
pp. C166-C180
Author(s):  
Mohamed Saleh Mehdi Al-Sultani ◽  
Igor Boglaev
Keyword(s):  
Numerical Solution ◽  
Iterative Methods ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Coupled Systems ◽  
Parabolic Problems ◽  
Nonlinear Elliptic ◽  
Monotone Iterations ◽  
Nonlinear Parabolic ◽  
Appl Math

The article deals with numerical methods for solving a coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing. We employ block monotone iterative methods based on the Jacobi and Gauss–Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of a solution are discussed. Numerical experiments are presented. References Al-Sultani, M. and Boglaev, I. ''Numerical solution of nonlinear elliptic systems by block monotone iterations''. ANZIAM J. 60:C79–C94, 2019. doi:10.21914/anziamj.v60i0.13986 Al-Sultani, M. ''Numerical solution of nonlinear parabolic systems by block monotone iterations''. Tech. Report, 2019. https://arxiv.org/abs/1905.03599 Boglaev, I. ''Inexact block monotone methods for solving nonlinear elliptic problems'' J. Comput. Appl. Math. 269:109–117, 2014. doi:10.1016/j.cam.2014.03.029 Lui, S. H. ''On monotone iteration and Schwarz methods for nonlinear parabolic PDEs''. J. Comput. Appl. Math. 161:449–468, 2003. doi:doi.org/10.1016/j.cam.2003.06.001 Pao, C. V. Nonlinear parabolic and elliptic equations. Plenum Press, New York, 1992. doi:10.1007/s002110050168 Pao C. V. ''Numerical analysis of coupled systems of nonlinear parabolic equations''. SIAM J. Numer. Anal. 36:393–416, 1999. doi:10.1137/S0036142996313166 Varga, R. S. Matrix iterative analysis. Springer, Berlin, 2000. 10.1007/978-3-642-05156-2 Zhao, Y. Numerical solutions of nonlinear parabolic problems using combined-block iterative methods. Masters Thesis, University of North Carolina, 2003. http://dl.uncw.edu/Etd/2003/zhaoy/yaxizhao.pdf

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