NMDP070723

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini July 7th 2023 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Consider the problem ( −∆u + 2u = 0 in Ω = (0 , 1)2, u = ex+y on ∂Ω, where the exact solution is uex = ex+y. 1. ( 2 pts) Write the weak formulation of the problem. 2. ( 4 pts) Prove that the hypothesis of the Lax-Milgram Lemma are satisfied. 3. ( 2 pts) Solve the problem with P1 triangular elements using Firedrake. 4. ( 2 pts) What are the expected orders of convergence in the H 1 and L2 norms with respect to the grid size. Motivate the answer. 5. ( 2 pts) Verify that the theoretical orders are confirmed by numerical computations. Exercise 2 (12 points) Consider the advection-diffusion-reaction problem  −∆u + ∇ · (bu) + u = 0, in Ω u = ϕ on ∂Ω, (1) where Ω = (0, 1) × (0, 1), b = (1000, 1000)T and: ϕ =    1 for x = 0, 0 < y < 1, 1 for y = 0, 0 < x < 1, 0 elsewhere . (2) 1. ( 3 pts) Is it possible to give an estimate about the maximum value of h that can be used when solving the problem with the classical Galerkin scheme ? 2. ( 3 pts) Write the weak formulation of the problem. 3. ( 3 pts) Solve the problem using the artificial viscosity method. 4. ( 3 pts) Solve the problem using the SUPG method. Exercise 3 (9 points) Consider the following method: un+1 j = un−1 j − a∆t ∆x (un j+1 − un j−1) (3) for solving the advection equation: ∂u ∂t + a ∂u ∂x = 0, with t > 0, x ∈ R, (4) and a > 0. 1. ( 2 pts) Draw the stencil of the method. 2. ( 7 pts) Prove that the method is second order accurate in both space and time. 1