Lesson 4 of 8 · Computational fluid dynamics
Adaptive mesh refinement principles
How this unit was checked 16 of 16 code examples ran · technical review corrected 1
Core theory
A uniform mesh wastes cells where the flow is calm and starves the wake where it isn’t. Adaptive refinement splits only the cells where the solution changes fastest. For incompressible flow we solve
(1)∇ · u = 0
(2)∂u∂t + (u · ∇)u = −1ρ∇p + ν∇2u
On each cell K of size hK we estimate the local error with a vorticity indicator, then flag the worst cells:
(3)ηK = hK ‖∇ω‖K, refine K if ηK > θ · maxJ ηJ
Halving h on a scheme of order p cuts the error by about 2p, which is why refining the right fraction of cells can beat refining all of them:
(4)eh ≈ C hp ⇒ eheh/2 ≈ 2p