Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Five point stencil — where it appears

2 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
-0.0833-0.167-0.0833-0.1671-0.167-0.0833-0.167-0.0833R · A · P, normalised to a unit centrestored entries per rowlevel 0 · 31×314.87/rowlevel 1 · 15×158.22/rowlevel 2 · 7×77.37/rowlevel 3 · 3×35.44/rowlevel 4 · 1×11.00/rowstill a stencil, still annihilates a constantentries in an interior row9weight outside the 3×30row sum0the isotropic model problemnine, at every level below the first

The 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.

iterative · multigrid θx (frequency across x)θy0π/2π0π/2πthe coarse grid'sunder 0.2under 0.40under 0.60under 0.80under 0.95under 1.01damping per sweeptwo routessmoothing factor, scanned1closed form130×30 frequency cellsthe marker is the mode nothing removes

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.

iterative · anisotropy

Named alongside it

The objects these essays reach for when they reach for this one.

Condition numberFourier modesGalerkin coarse operatorSmoothing factorThe V-cycleAnisotropyGrid complexityProlongationRestrictionWeighted jacobi

All concepts