NMDP210623

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini June 21st 2023 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Consider the problem    −∆u = f in Ω = (0, 1)2, ∇u · n = gN on ΓN = {(x, y) ∈ R2|x = 0}, u = 0 on Γ D = ∂Ω \ ΓN , where the exact solution is u = y(1 − y) cos  πx 2  . 1. ( 2 pts) Write the weak formulation of the problem. 2. ( 1 pts) Compute f and gN from the exact solution and verify that the Dirichlet boundary condition is satisfied. 3. ( 2 pts) Solve 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. ( 2pts) Suppose that the problem is solved with the same boundary conditions, but a different source term ef. Call the corresponding solution eu. Write the strong form of the problem that must be solved to determine w := u −eu. 6. ( 3 pts) Using the Poincar´ e inequality, prove that there exists a constantC > 0 such that ∥w∥L2(Ω) ≤ C∥f − ef ∥L2(Ω). Exercise 2 (12 points) Consider the following Stokes problem  −ν∆u + ∇p = f , in Ω ∇ · u = 0, (1) where Ω ⊂ R2 and u = g on Γ = ∂Ω. 1. ( 2 pts) Prove that g must satisfy the compatibility condition R Γ g · ndγ = 0. 2. ( 2 pts) Let Ω be the unit square [0 , 1] × [0, 1] in xy plane, f = 0 and g = [gx, gy] with gx =  0 for y = 0, y = 1, 0 < x < 1, y − y2 for x = 0, x = 1, 0 < y < 1, gy = 0. (2) Verify that R Γ g · ndγ = 0. 3. ( 3 pts) Find (analitically) a solution in the form u = [ux, uy] where uy = 0. 4. ( 2 pts) Write the weak formulation of (1). 5. ( 3 pts) Solve the problem with Firedrake using P2 finite elements for the velocity and P1 for the pressure with h = 0.05, 0.025, 0.0125 and ν = 1. Comment on this choice of the discrete spaces. Exercise 3 (9…