Multigrid and smoothing properties

For the uniform one-dimensional Poisson matrix, a weighted-Jacobi update applies the inverse diagonal to the residual with a weight omega. Its error iteration is identity minus omega D inverse A. On a periodic mean-zero grid, or in local Fourier analysis away from boundaries, the eigenvalue on a frequency theta is one minus omega times(1−cos theta). The constant periodic mode must be removed because Poisson has a nullspace there. A separate SPD boundary-value system can then be used for the coarse-projector discussion.

With omega=2/3, frequencies from pi/2 to pi have amplification between−1/3 and 1/3, so one sweep damps that high-frequency band by at least a factor three in magnitude. Low frequencies have multipliers approaching one as the mesh is refined. Ordinary unweighted Jacobi at omega=1 gives multiplier−1 at theta=pi and does not damp that highest periodic mode at all. Merely naming Jacobi without its damping weight can therefore obscure the smoothing property the multigrid argument needs.

For an SPD fine-grid matrix A and a full-column-rank prolongation P, define the Galerkin coarse matrix as P transpose A P. Solve its residual equation exactly and prolong the correction. The resulting fine-grid error operator is identity minus P times the coarse inverse times P transpose A. This is the A-orthogonal projector onto the complement of the coarse space, so it removes errors in the range of P and cannot increase the A-energy norm. It does not remove arbitrary fine-grid error in one step.

The coarse space must approximate what the smoother leaves behind. For a regular elliptic problem with appropriate nested grids and transfers, smooth fine-grid errors can often be represented cheaply on the next level. Strong anisotropy or discontinuous coefficients may violate this simple picture and require line smoothers, altered interpolation or another coarse space. An exact Galerkin solve supplies a clean projector identity, but an approximate recursive coarse solve adds another error operator that belongs in the full analysis.

The two-grid error propagator combines presmoothing, coarse correction and postsmoothing in their actual order. A bound strictly below one in the energy norm, uniform over mesh sizes, yields mesh-independent contraction. Obtaining that bound requires smoothing and approximation estimates that complement each other; neither the high-frequency factor nor the coarse projector’s nonexpansiveness alone gives it. The same care is needed when recursive coarse solves are used to form V or W cycles.

Work estimates also have assumptions. For geometrically shrinking grids and a bounded amount of work per degree of freedom per level, one cycle can cost order the number of fine-grid unknowns. A uniform contraction then leads to an efficient solve to a stated tolerance, but a smoother that stalls or a poor coarse space can erase that advantage. Report the norm, problem class, transfers and stopping rule alongside timing claims. The worked calculation verifies smoothing of selected modes; it intentionally does not convert three favorable multipliers into a theorem for an arbitrary multigrid hierarchy.