Esame 240624

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini June 24th 2024 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Consider the problem ( −∆u − ru = f in Ω = (0, 1) × (0, 1) u = gD on ∂Ω (1) with r > 0. 1. ( 2 pts) Write the weak formulation of (1). 2. ( 2 pts) Given the exact solution u(x, y) = sin(πx) sin(2πy) + x, compute f and gD. 3. ( 3 pts) Solve the problem using Firedrake and linear finite elements, with r = 1/(4π2). 4. ( 2 pts) Compute and comment the error between the exact and the numerical solution in the H 1 norm for different values of mesh size. 5. ( 3 pts) Prove that the problem is well posed under suitable conditions on the value of r. Exercise 2 (11 points) Consider the unsteady Stokes problem    ∂u ∂t − ∆u + ∇p = f in Ω × (0, T] ∇ · u = 0 in Ω × (0, T] u = gD on ∂Ω × (0, T] u = 0 in Ω × {t = 0} (2) with Ω = (0, 1)2, f = [0, −500], gD = [sin(2πt), 0] on y = 1, gD = [0, 0] on ∂Ω \ {y = 1}. 1. ( 1 pts) Write the semi-discretization in time of (2) using the implicit Euler method. 2. ( 2 pts) Write the weak form of the semi-discretized problem. 3. ( 4 pts) Write the algebraic form of the problem and the corresponding block system. 4. ( 4 pts) Solve the problem using Firedrake, with T = 1 and dt = 0.1 and a suitable pair of finite elements for velocity and pressure. Plot velocity and pressure at each time step. Exercise 3 (10 points) Consider the following linear advection equation ∂u ∂t + 2∂u ∂x = 0, x ∈ (0, 25), t ∈ [0, 8] (3) with periodic boundary conditions and the initial condition u0(x) = e−20(x−2)2 + e−(x−5)2 . 1. ( 1 pts) Derive the exact solution of the problem (for the unbounded domain). 2. ( 3 pts) Use the code upwind.py (that implements the upwind method) to solve (3) using ∆ x = 0.05…