NMDP060224

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini February 6th 2024 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Let Ω = (0, 2) × (0, 2), γ1 = 2 × (0, 2) and γ2 = ∂Ω \ γ1. Consider the following problem in Ω − ∂2u ∂x2 − 2 ∂2u ∂y 2 = 1, (1) with ∂u ∂x = 0 on γ1 and u = 0 on γ2. 1. ( 2 pts) Write (1) in the form −∇ · (K∇u) = f where K is a 2 by 2 matrix. 2. ( 2 pts) Write the weak formulation of the form derived in point 1. 3. ( 2 pts) Write the discrete weak formulation. 4. ( 3 pts) Solve the problem using Firedrake and quadratic finite element. (A generic 2x2 matrix K can be built using command K = as matrix([[a, b],[c, d]]) and multiplied by a vector v using dot(K, v) ). 5. ( 3 pts) Prove that the problem is well-posed. Exercise 2 (12 points) Consider the Stokes problem    −∆u + ∇p = f in Ω = (0, 4) × (0, 1), ∇ · u = 0, u = gD on Γ1, ∂u ∂n − pn = gN on Γ2, u = 0 on Γ 3 ∪ Γ4, (2) with f = [sin θ, − cos θ], describing viscous flow down a channel with angle θ (see the figure below). The exact solution for velocity is u(x, y) =  a − sin θ 2  y(y − 1), 0  . x y θ x y Γ1 Γ3 Γ2 Γ4 1. ( 2 pts) Compute the pressure field p(x, y), imposing p(0, 0) = 0. 2. ( 2 pts) Compute boundary data gD(y) and gN(y) from the exact solution. 3. ( 2 pts) Write the weak formulation of the problem. 4. ( 3 pts) Solve the problem with P1 − P1 elements using Firedrake and setting a = 0 .1, θ = π/6. (Build the mesh using command mesh = RectangleMesh(20, 10, 4., 1., diagonal=’crossed’) ; use command n = FacetNormal(mesh) to obtain the normal to the boundary.) 5. ( 3 pts) Plot the velocity and pressure fields and comment the result. Describe some ways to fix the issue on the pressure. Is refining the mesh one of them? 1 Exercise 3 (9…