u/w142236

Also, this is a pizza 👉🍕

This lower level (around the 1km elevation mark) layer of unusually dry air has been hanging over San Diego for several days now and the minimum dewpoint temp has been steadily dropping over time.

Here’s a breakdown of the images:

- First image: Formula relating exact V\_h and the exact potentials which create it, formulas for nth-order approximations of the potentials, the goal, and a flowchart. Also, I forgot to include the residual formulas is δ**V**_h^(n) = **V**_h - **V**_h^(n)

Images 2-5 are the flowchart and relevant formulas expanded with the boundary conditions explicitly defined

- Second image: 0th-order approximation problem for psi_h with inhomogeneous Dirichlet boundary condition defined

- Third image: 0th-order approximation problem for chi\_h with inhomogeneous Dirichlet boundary condition defined with indication that these boundary conditions depend on 0th-order psi_h

- Fourth image: 0th-order approximation of Vh is computed using 0th-order approximation solutions of potentials, then 1st-order residual of **V**_h is computed, 1st-order correction of psi\_h problem is setup using 1st-order residual to determine boundary conditions, and 1st-order correction of chi_h problem is setup using 1st-order residual and 1st-order correction of psi_h solution to determine boundary conditions.

The general form for the nth-order boundary correction formulas for the nth-order potential correction problems are

1. Δψ_h^(n) (s) = Δψ_h^(n) (s_0) - ∫\_{s_0}^s [δV_h^(n) • **n**]d.s.

• Δψ\_h\^(n) (0,0) = 0

2. Δχ_h^(n) (s) = Δχ_h^(n) (s_0) + ∫_{s_0}^s [δV_h^(n) • s]ds - ∫_{s_0}^s [∇(Δχ_h^(n) ) • **n**]ds

• Δχ_h^(n) (0,0) = 0

- Fifth image: 1st-order approximations of psi_h and chi_h are computed using 1st-order corrections, these first-order potentials have then been used to construct the 1st-order approximation of **V**_h, and the cycle becomes visible as we compute 2nd-order corrections from the 2nd-order residuals (using the definitions for the nth-order boundary corrections), then 2nd-order approximations of potentials are obtained and used to determine 2nd-order approximation of **V**_h, and then lastly that is used to obtain the 3rd-order residual of **V**_h.

This cycle of computing nth-order corrections, nth-order approximations, and (n+1)th-order residuals continues until we arbitrarily decide to stop.

___

I want to know if there’s a name for this repeated residual-correction process used to refine a solution. Maybe what fields you might have seen it in as well.

Supposedly this process reduces the error of the approximation of **V**_h with each iteration, however when I tried it symbolically in Python, my error actually increased after 1 or 2 iterations, and I don’t know why. So, if I could just get a name for this process, I can more easily find literature into the methodology, and that will help me figure out where I went wrong.

If anyone would like, I can also provide my Python code.

Was just a YouTube thumbnail I saw. Thought it deserved a laugh.