Error estimates of the finite volume scheme for the nonlinear tensor-driven anisotropic diffusion

Authors:
Olga Drblíková;Angela Handlovičová;Karol Mikula
Affiliations:
Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 81368 Bratislava, Slovakia;Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 81368 Bratislava, Slovakia;Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 81368 Bratislava, Slovakia
Venue:
Applied Numerical Mathematics
Year:
2009

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Abstract

The paper deals with an error analysis of the semi-implicit diamond-cell finite volume scheme, introduced in [O. Drblikova, K. Mikula, Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (1) (2007) 37-60], for solving the nonlinear tensor-driven anisotropic diffusion. First we present the finite volume scheme and its basic properties. Then the error estimate analysis is presented, where the piecewise constant approximation given by the finite volume scheme is compared with the weak solution to the problem. We proved that the error of the approximate solution in L^2-norm is of order h, where h is a spatial resolution step under the natural relation k~h^2, where k is a time discretization step. The numerical results devoted to image processing applications are also given.