Assignment
Consider the following 3D equations defined on Ω = (—1, 1) × (—1, 1) × (—1, 1):
The boundary conditions are
F1 (—1,y, z) = Fb (y, z), F3 (x,—1, z) = Fb (x,z), F5 (x,y,—1) = Fb (x,y),
F2 (1,y, z) = F4 (x,1, z) = F6 (x,y, 1) = 0,
where
• Solve this system of equations numerically for σ = 0.1, 1, 10, 100.
• Let m = F1 + F2 + F3 + F4 + F5 + F6 . Assume σ = 1/ϵ . Derive the equation of m that approximates the system of F1 , ··· , F6 up to the first order.
Write a report including
1. An introduction to the numerical method used to solve the problem.
2. The numerical settings.
3. Numerical solutions represented by figures.
4. Plots showing the convergence of the iterative method.
5. Derivation of the first-order approximation.
Submit your report and your codes to Canvas no later than 4 Apr 2025.