I know (from https://scicomp.stackexchange.com/a/31339/20545, among others) that I need a certain mesh density in FEM, else I might get non-physical oscillations in my solution.
How do multigrid approaches deal with such problems? They reduce the problem to a coarser mesh, solve it there, and use the result (by smoothening) for either solving the problem directly on the fine mesh, or for preconditioning. While solving on the coarse mesh, those methods will result in oscillations. Will those be smoothened out during the process back to the fine mesh, or are there other restrictions which avoid bringing those oscillations back to the fine mesh?
Or is my understanding of the method wrong?
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The point important to understand when thinking about multigrid is that the lower levels of the hierarchy do not actually have to solve the problem accurately. Rather, the operators at the lower levels just need to provide good approximations of a part of the spectrum (eigenvalues) of the operator on the finest level -- specifically, they need to well approximate the low-frequency component of the solution. (The smoother takes care of the high frequency component.)
As a consequence, the "qualitative properties" of the solution on the lower levels don't actually matter very much.
answered Aug 1, 2019 at 2:50
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$\begingroup$ If you apply standard galerkin to a problem, which then generates oscillations on coarse meshes, is it even possible to construct a smoothing iteration scheme ? $\endgroup$– cfdlabAug 1, 2019 at 14:18
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1$\begingroup$ @cpraveen: That's the wrong question. In multigrid methods, you solve the equations on a coarser level where the (restricted, smoothed) residual acts as the right hand side. You will then get a solution that represents the error on this level. It may very well be oscillatory, but what you use as correction on the next finer level is what you get after (i) you transfer this error representation to the next finer level, (ii) you smooth it. Smoothing here is meant in the common sense of the word: In the simplest case, you just average neighboring nodes. So yes, it is possible :-) $\endgroup$ Aug 1, 2019 at 21:56
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$\begingroup$ I was considering a Galerkin method applied to a convection-diffusion problem. My doubt was about smoothing properties of the iterative scheme. Suppose $h$ is finest mesh on which mesh Peclet < 2 so $A_h$ is a good matrix (M-matrix?). Suppose on $2h$ mesh, mesh Peclet > 2 so that $A_{2h}$ is not an M-matrix. Would the iterations using $A_{2h}$ actually reduce the high frequency errors on $2h$-mesh ? @wolfgang-bangerth $\endgroup$– cfdlabAug 2, 2019 at 10:13
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$\begingroup$ Ah, I see -- you might have to stabilize on the coarser meshes as appropriate. But I really don't know. $\endgroup$ Aug 4, 2019 at 11:30