NMDP040923

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini September 4th 2023 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Consider the problem ( −∇ · (µ∇u) = f in Ω = (0, 1)2, u = 0 on ∂Ω, with f(x, y) = 1 and µ(x, y) = 1 + 5y. 1. ( 2 pts) Write the weak formulation of the problem. 2. ( 2 pts) Prove that the weak problem is well posed. 3. ( 2 pts) Compute a numerical solution uh of the problem with P1 triangular elements using Firedrake. 4. ( 2 pts) Verify the order of convergence in the H1 norm with respect to the grid size. 5. ( 4 pts) Solve again the problem with uniform diffusivity µ = 1; call the solution wh. Compute numerically (using the Firedrake command assemble) the quantities F (uh) and F (wh), where F (v) = 1 2 Z Ω ∇v · ∇v dΩ − Z Ω f vdΩ. Compare the results and comment them. Exercise 2 (12 points) Consider the following Stokes problem    −ν∆u + ∇p = f in Ω = (0, 1)2 ∇ · u = 0, in Ω = (0, 1)2 u = g, on ∂Ω. (1) 1. ( 3 pts) Consider f = [0 , −y] and the exact velocity solution uex = [ x2 + y2, −2xy]. Verify that uex is divergence-free and compute the corresponding zero-mean pressure solution pex. 2. ( 2 pts) Write the weak formulation of (1). 3. ( 3 pts) Solve the problem with Firedrake using P2 finite elements for the velocity and P1 finite elements for the pressure. 4. ( 2 pts) Verify the order of convergence for velocity in the H1 norm with respect to the grid size. 5. ( 2 pts) Solve the problem with Firedrake using P1 finite elements for the velocity and P1 and the penalty stabilization method. Exercise 3 (9 points) Consider the following hyperbolic equation ∂u ∂t + a ∂u ∂x = 0, (2) where t > 0, x ∈ R and a is a real positive number. We want to devise an explicit numerical method based on finite…