Local Fourier analysis(LFA)is a useful tool in predicting the convergence factors of geometric multigrid methods(GMG).As is well known,on rectangular domains with periodic boundary conditions this analysis gives the e...Local Fourier analysis(LFA)is a useful tool in predicting the convergence factors of geometric multigrid methods(GMG).As is well known,on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such met hods.When other boundary conditions are considered,however,this analysis was judged as been heuristic,with limited capabilities in predicting multigrid convergence rates.In this work,using the Fourier method,we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems,some of which can not be handled by the traditional rigorous Fourier analysis.展开更多
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 acc...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 whichε>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 forε>>1.In this sense,an xy-zebra-GS smoother was proposed,which proved to be efficient and robust for the different anisotropy coefficients.Moreover,the convergence factors calculated empirically and by LFA are in agreement.展开更多
文摘Local Fourier analysis(LFA)is a useful tool in predicting the convergence factors of geometric multigrid methods(GMG).As is well known,on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such met hods.When other boundary conditions are considered,however,this analysis was judged as been heuristic,with limited capabilities in predicting multigrid convergence rates.In this work,using the Fourier method,we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems,some of which can not be handled by the traditional rigorous Fourier analysis.
文摘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 whichε>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 forε>>1.In this sense,an xy-zebra-GS smoother was proposed,which proved to be efficient and robust for the different anisotropy coefficients.Moreover,the convergence factors calculated empirically and by LFA are in agreement.
