Lab assignment 1: Due Friday April 3, 18:00 EDT

For this assignment, you will need a set up to work with Python code, in particular, to develop Jupyter notebooks, as well as a GitHub account to share your notebooks.

We will solve the following boundary value problem by a finite element method. $$ \begin{cases} -\Delta u = f&\textrm{in}\,\,\Omega,\\ u = g &\textrm{on}\,\,\partial\Omega. \end{cases}\qquad\qquad(*) $$

Part 1: Implementation

• Taking inspiration from the reference, implement an affine finite element method for the Poisson equation in 2D.
• Your code should be able to handle at least one convex domain (such as a square or a triangle), and a domain with a reentrant corner (such as an L-shaped domain).
• It should also be able to handle inhomogeneous Dirichlet boundary conditions.
• Implement either the red refinement or a bisection procedure for mesh subdivision.

Part 2: Numerical experiments

1. Pick a convex domain $\Omega$ and a smooth function $u$, and compute $f=-\Delta u$ in $\Omega$. Then solve (*) with $f$ just computed and with $g=u|_{\partial\Omega}$, on several grids with decreasing meshsize.
2. Plot the $H^1$-error of the finite element solution against the meshsize $h$ on a logarithmic scale, and estimate the convergence rate $\alpha$ in $error\sim h^\alpha$ from the graph. Is it consistent with theory?
3. Repeat the preceding exercise for the $L^2$-error of the finite element solution. Explain the results.
4. Repeat the preceding 3 exercises for a smooth function $u$ on a domain with a reentrant corner. Explain the results.
5. In a domain with a reentrant corner, consider a solution of the form $u(r,\theta)=r^a\sin(a\theta)$, where the coordinate origin is at the reentrant corner, and the rays $\theta=0$ and $\theta=\pi/a$ correspond to the two sides of the corner. For example, for an L-shaped domain, we would have $a=\frac23$. This is the function from Assignment 1, Problem 5. Repeat exercises 1–3. If your code allows, experiment with a couple of different values of $a$, that is, with different values of the reentrant angle. Explain the results.

How and what to submit

• Include theory and commentaries in your Jupyter notebooks, by using Markdown cells.
• Notebooks without sufficient commentaries will not be accepted.
• You can update your notebooks up to the time limit, but please do not commit any overdue updates.

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