Lane-Emden Equations and Related Topics in Nonlinear Elliptic and Parabolic Problems

In this paper we consider the error estimates for some boundary-flux calculation procedures applied to two-point semilinear and strongly nonlinear elliptic boundary value problems. The case of semilinear parabolic problems is also studied. We prove that the computed flux is superconvergent with second and third order of convergence for linear and quadratic elements respectively. Corresponding estimates for higher order elements may also be obtained by following the general line of argument.

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On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2013-0011 ◽

2013 ◽

Vol 13 (4) ◽

pp. 369-410 ◽

Cited By ~ 7

Author(s):

Boris Andreianov ◽

Mostafa Bendahmane ◽

Florence Hubert

Keyword(s):

Functional Analysis ◽

Finite Volume ◽

Parabolic Problems ◽

Sobolev Embedding ◽

Finite Volume Schemes ◽

Compactness Result ◽

Nonlinear Elliptic ◽

Analysis Tools ◽

Consistency Results ◽

Discrete Compactness

Abstract. We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W1,p compactness, discrete compactness in space and in time) for the so-called Discrete Duality Finite Volume (DDFV) schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [IMA J. Numer. Anal., 32 (2012), pp. 1574–1603]. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov (1969); others generalize the ideas known for the 2D DDFV schemes or for traditional two-point-flux finite volume schemes. We illustrate the use of these tools by studying convergence of discretizations of nonlinear elliptic-parabolic problems of Leray–Lions kind, and provide numerical results for this example.

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Block monotone iterations for solving coupled systems of nonlinear parabolic equations

ANZIAM Journal ◽

10.21914/anziamj.v61i0.15144 ◽

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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Monotone schemes for a class of nonlinear elliptic and parabolic problems

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2008.09.018 ◽

2010 ◽

Vol 11 (6) ◽

pp. 4546-4553

Author(s):

N. Limić ◽

M. Rogina

Keyword(s):

Parabolic Problems ◽

Monotone Schemes ◽

Nonlinear Elliptic

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

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202513500358 ◽

2013 ◽

Vol 23 (13) ◽

pp. 2395-2432 ◽

Cited By ~ 63

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.

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