Question

Exercise 2: Finite element method We are interested in computing numerically the solution to a 2D Laplace equation u 0, The triangulated domain is given in the file mesh.mat on Blackboard. which contains the V × 2 nnatrix vertices storing the two coordinates of the vertices and a F × 3 matrix triangles in which each ro w J contains the indices in {1,····V) of the three vertices of the j-th triangle. a) Using for example MATLAB's triplot or trimesh function, plot and show the triangulated domain. How can you characterize edges that belong to the boundary of the domain? Write a code to find al the indices of boundary vertices and check it by plotting those points on top of the initial mesh. b) we will take the boundary conditions u = q where the function q is equal to 0 on the exterior square,1 on the small inner circle and 1 on the large inner circle. For this example, propose a simple way to distinguish these three different sets of boundary vertices and compute the corresponding vector of boundary values. c) From now on, we consider a Pl finite element method to solve the PDE. What are the unkowns and the size of the associated linear system in this case? With the notations of the class, how does the right hand side vector b write? Based on the expressions of Lecture 6, compute numerically this vector b d) In the same way, write a code to compute the stiffness matrix associated to the problem. Give its conditioning number e Deduce the numerical values of the solution u and plot it using trimesh. Bonus question: For the same triangulated domain, we now consider Poissons equation Δu f such that u = 0 on the boundary. The values of the function f at the vertices are given by the vector f in mesh.mat. Using again a Pl finite element method for this problem, how does the resulting linear system change compared to c)? Write a code to compute the numerical value of b using, on each triangle T, the following simple quadrature formula: g(r)dr ~ Area(T) × g(n) + g(g) + g(m where v, v2,s are the three vertices of T. Compute and plot the solution u

Answer 1

Neglecting the atmospheric pressure

Density of mercury = 0.490 lb/in3

gravity = 32.17 ft/s2

h = 15 inch

Pressure in water main = pgh

= 0.490x32.17x12x15

=2837.4 psi