smoothing-square
At its defaults it draws damping over the frequency square, ε = 1, point relaxation. A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
smoothing-square is one function in lib/figures/grid2d.js —
two dimensions — where the galerkin identity stops being an identity. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
eps: 0.001
The arguments are the ones A direction the smoother cannot see passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
eps: 0.1
The arguments are the ones A direction the smoother cannot see passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
eps: 1
The arguments are the ones An answer that is known passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
eps: 0.001, coarsening: "semi-y"
The arguments are the ones Coarsening in one direction only passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
eps: 0.001, kind: "liney"
The arguments are the ones Smoothing a whole line at once passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A square of frequency pairs shaded by how much one relaxation sweep damps each mode. The lower-left quarter, which the coarse grid represents, is outlined. A marker sits on the least-damped mode outside it.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
5 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
the smoothing factor at ε = 1, scanned and in closed form agree — asserted 3 times
and the cell grid agrees with the fine scan
the scan found a surviving mode
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A direction the smoother cannot see
Give the Laplacian a strong direction and multigrid stops working — from 0.2016 a cycle to 0.9565 — with every component unchanged and the condition number identical to twelve digits. The problem did not get harder. The link between the method's two halves broke.
Two errors, and whose fault they areAn answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
Iterating, instead of factorisingCoarsening in one direction only
Leave the smoother alone and halve only the strong direction, and the smoothing factor is 0.3340 — identical to line relaxation's, at every anisotropy and every weight, to twelve digits. The convergence factors are then a factor of three apart, and at 45° both repairs fail outright.
Iterating, instead of factorisingSmoothing a whole line at once
Solve every grid line in the strong direction exactly rather than sweeping over it, and the smoothing factor goes from 0.9993 back to 0.3340 — which is the one-dimensional answer, on a problem that is not one-dimensional. The repair replaces one ε in the closed form by a one.
Iterating, instead of factorisingThe coarse problem is a different problem
In one dimension the Galerkin coarse operator is the coarse discretisation, entry for entry — this site asserted it. In two dimensions a five-point operator produces a nine-point coarse one, so the recursion solves a different discretisation at every level below the first, and converges at 0.20 a cycle regardless.
Iterating, instead of factorisingThe error smoothing cannot reach
One weighted Jacobi sweep multiplies every mode of the error by a number, and the number is a sine. Half the modes are cut by three or better, and the other half come back at 0.999 — which is not a failure of the method but the fact the whole of multigrid is built on.