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# On the Robustness of the xy-Zebra-Gauss-Seidel Smoother on an Anisotropic Diffusion Problem

Michely Laís de Oliveira1,*, Marcio Augusto Villela Pinto2, Simone de Fátima Tomazzoni Gonçalves2, Grazielli Vassoler Rutz3

1 Graduate Program in Numerical Methods in Engineering, Federal University of Paraná, 81531-980, Curitiba-PR, Brazil.
2 Department of Mechanical Engineering, Federal University of Paraná, 81531-980, Curitiba-PR, Brazil.
3 Federal Institute of Education, Science and Technology of Santa Catarina, 89813-000, Chapecó-SC, Brazil.
* Corresponding Author: Michely Laís de Oliveira. Email: michelylais@ufpr.br.

Computer Modeling in Engineering & Sciences 2018, 117(2), 251-270. https://doi.org/10.31614/cmes.2018.04237

## Abstract

Studies of problems involving physical anisotropy are applied in sciences and engineering, for instance, when the thermal conductivity depends on the direction. In this study, the multigrid method was used in order to accelerate the convergence of the iterative methods used to solve this type of problem. The asymptotic convergence factor of the multigrid was determined empirically (computer aided) and also by employing local Fourier analysis (LFA). The mathematical model studied was the 2D anisotropic diffusion equation, in which ε > 0 was the coefficient of a nisotropy. The equation was discretized by the Finite Difference Method (FDM) and Central Differencing Scheme (CDS). Correction Scheme (CS), pointwise Gauss-Seidel smoothers (Lexicographic and Red-Black ordering), and line Gauss-Seidel smoothers (Lexicographic and Zebra ordering) in x and y directions were used for building the multigrid. The best asymptotic convergence factor was obtained by the Gauss-Seidel method in the direction x for 0 < ε << 1 and in the direction y for ε >> 1. In this sense, an xy-zebra-GS smoother was proposed, which proved to be efficient and robust for the different anisotropy coefficients. Moreover, t h e c o nvergence factors calculated empirically and by LFA are in agreement.

## Keywords

Laís, M., Augusto, M., De, S., Rutz, G. V. (2018). On the Robustness of the xy-Zebra-Gauss-Seidel Smoother on an Anisotropic Diffusion Problem. CMES-Computer Modeling in Engineering & Sciences, 117(2), 251–270.

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