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Abstract
In the present work certain porous medium equations like linear diffusion, linear convection dominated and nonlinear diffusion problems which appear as parabolic partial differential equations (PDE’s) are considered. The numerical solution of linear problems is carried out by establishing a fully discretization scheme which is partitioned to spatial discretization by using finite element method based upon standard or Petrov-Galerkin basis functions and time discretization by using Rothe method (method of lines). On the other hand the nonlinear problems are first linearized by the use of some relaxation functions and after that are solved as illustrated before. There are some difficulties introduced by the “nonlinear” character of the equation and by its degenerate nature. To attack the first difficulty we use a certain linearization scheme, at which a “nonstandard time discretization with a relaxation function”, to transform our problem to a linear diffusion one. The problem of degeneracy is treated with the regularization of the nonlinear functions.
Keywords: finite element method, Galerkin method, Petrov-Galerkin method, Rothe method, error estimates, linear parabolic partial differential equation, nonlinear parabolic partial differential equation, porous medium, functional analysis, variational analysis.