Problem set 3: Finite element methods#

Note: If you miss the problem solving class, you have to submit solutions to all the exercises below in Studium before the deadline.

Exercise 1#

Consider the following boundary-value problem:

\[\begin{split} \begin{array}{rl} -\epsilon u'' + x u' + u = 1, & x \in \Omega = [0, 1] \\ u(0) = 1, & \\ u'(1) = 7, & \end{array} \end{split}\]

where \(\epsilon > 0\) is a constant.

a) Derive a weak form of the problem, with appropriate spaces.

b) Explain the difference between test and trial spaces in general. What are the test/trial spaces in a)?

c) Sketch the five “hat functions” that form a basis for the space of piecewise linear functions on a mesh of four subintervals of equal length.

d) The hat function \(\varphi_1\) satisfies \(\varphi_1(0.25) = 1\). How does \(\varphi_1\) depend on \(x\)? Write down the mathematical expression.

e) For the mesh of four equal subintervals, formulate the finite element method and derive the corresponding linear system of equations. You do not need to evaluate any of the integrals that appear in the stiffness matrix and load vector.