u/w142236

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.

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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.