Regularity Estimates for Nonlinear Elliptic and Parabolic Problems

2012 ◽  
Author(s):  
John Lewis ◽  
Peter Lindqvist ◽  
Juan J. Manfredi ◽  
Sandro Salsa
Keyword(s):  
Parabolic Problems ◽  
Nonlinear Elliptic ◽  
Regularity Estimates

Regularity estimates for nonlinear elliptic measure data problems with nonstandard growth

2019 ◽  
Vol 182 ◽  
pp. 303-315 ◽  
Author(s):  
Sun-Sig Byun ◽  
Shuang Liang ◽  
Yeonghun Youn
Keyword(s):  
Measure Data ◽  
Nonstandard Growth ◽  
Nonlinear Elliptic ◽  
Regularity Estimates

Weighted Orlicz regularity estimates for fully nonlinear elliptic equations with asymptotic convexity

2019 ◽  
Vol 21 (04) ◽  
pp. 1850024 ◽  
Author(s):  
Mikyoung Lee
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Operator ◽  
Orlicz Spaces ◽  
Nonlinear Elliptic Equations ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Regularity Estimates ◽  
Hessian Estimates ◽  
Asymptotic Convexity

We prove interior Hessian estimates in the setting of weighted Orlicz spaces for viscosity solutions of fully nonlinear, uniformly elliptic equations [Formula: see text] under asymptotic assumptions on the nonlinear operator [Formula: see text] The results are further extended to fully nonlinear, asymptotically elliptic equations.

scholarly journals Superconvergence analysis of flux computations for nonlinear problems

1989 ◽  
Vol 40 (3) ◽  
pp. 465-479 ◽  
Author(s):  
S.-S. Chow ◽  
R.D. Lazarov
Keyword(s):  
Elliptic Boundary ◽  
Nonlinear Problems ◽  
Parabolic Problems ◽  
Elliptic Boundary Value ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
General Line ◽  
Nonlinear Elliptic Boundary ◽  
Boundary Flux ◽  
Semilinear Parabolic

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.

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

2013 ◽  
Vol 13 (4) ◽  
pp. 369-410 ◽  
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.

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

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.

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