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000848186 1001_ $$0P:(DE-HGF)0$$aBolten, M.$$b0$$eCorresponding author
000848186 245__ $$aFourier Analysis of Periodic Stencils in Multigrid Methods
000848186 260__ $$aPhiladelphia, Pa.$$bSIAM$$c2018
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000848186 520__ $$aMany applications require the numerical solution of a partial differential equation (PDE), leading to large and sparse linear systems. Often a multigrid method can solve these systems efficiently. To adapt a multigrid method to a given problem, local Fourier analysis (LFA) can be used. It provides quantitative predictions about the behavior of the components of a multigrid method. In this paper we generalize LFA to handle what we call periodic stencils. An operator given by a periodic stencil has a block Fourier symbol representation. It gives a way to compute the spectral radius and norm of the operator. Furthermore block Fourier symbols can be used to find out how an operator acts on smooth/oscillatory input and whether its output will be smooth/oscillatory. This information can then be used to construct efficient smoothers and coarse grid corrections. We consider a particular PDE with jumping coefficients and show that it leads to a periodic stencil. LFA shows that the Jacobi method is a suitable smoother for this problem and an operator dependent interpolation is better than linear interpolation, as suggested by numerical experiments described in the literature. If an operator is given by an ordinary stencil, then block smoothers yield periodic stencils if the blocks correspond to rectangles in the domain. LFA shows that the block Jacobi and the red-black block Jacobi method efficiently reduce more frequencies than their pointwise versions. Further, it yields that a block smoother used in combination with aggressive coarsening can to some degree compensate for the reduced convergence rate caused by aggressive coarsening.
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000848186 7001_ $$0P:(DE-Juel1)174446$$aRittich, Hannah$$b1
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