coarse-stencil
At its defaults it draws the galerkin operator at level 1: nine points. A three-by-three arrangement of discs carrying the stencil's coefficients, with the zero positions drawn small, beside a bar chart of stored entries per row at each level of the hierarchy.
coarse-stencil 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 three-by-three arrangement of discs carrying the stencil's coefficients, with the zero positions drawn small, beside a bar chart of stored entries per row at each level of the hierarchy.
level: 1
The arguments are the ones Elimination is a sequence of choices 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 three-by-three arrangement of discs carrying the stencil's coefficients, with the zero positions drawn small, beside a bar chart of stored entries per row at each level of the hierarchy.
level: 0
The arguments are the ones The coarse problem is a different problem 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 three-by-three arrangement of discs carrying the stencil's coefficients, with the zero positions drawn small, beside a bar chart of stored entries per row at each level of the hierarchy.
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.
7 distinct claims across 3 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.
a level inside the hierarchy
and every coarse one has nine
and one the hierarchy reaches
the fine operator has five entries in an interior row
the grid size is one less than a power of two
the operator restricted to a 3×3 neighbourhood is the whole row
the stencil still annihilates a constant
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.
Elimination is a sequence of choices
Gaussian elimination is taught as a procedure with no decisions in it. There is one decision at every step — which row to use — and every stability property the algorithm has comes from making it well.
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.
Eigenvalues, singular values, rankThe form that makes it affordable
One Householder reduction, done once, turns every subsequent iteration of the eigenvalue algorithm from cubic to quadratic cost. It changes no answer at all, which is why it is easy to describe as an optimisation and wrong to.
Iterating, instead of factorisingThe same problem on a coarser grid
Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.