Singular Solutions of Nonlinear Elliptic and Parabolic Equations

2016 ◽  
Author(s):  
Alexander A. Kovalevsky ◽  
Igor I. Skrypnik ◽  
Andrey E. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Singular Solutions ◽  
Nonlinear Elliptic ◽  
Elliptic And Parabolic Equations

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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