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Implicit a posteriori error estimation using patch recovery techniques

100%
Horváth T., Izsák F.
Open Mathematics
|
2012
|
tom 10
|
nr 1
55-72
EN
We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging which define a family of implicit a posteriori error estimators. We will demonstrate the performance and the favor of the method through numerical experiments.
2
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A finite difference method for fractional diffusion equations with Neumann boundary conditions

100%
Szekeres B. J., Izsák F.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald–Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.
3
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An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

81%
Faragó I., Izsák F., Szabó T., Kriston Á.
Open Mathematics
|
2013
|
tom 11
|
nr 4
746-759
EN
An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.
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