2 minute read

Homework?

As an educator, one of our duty is to do our best to let the students learn (outcomes). However, circa two years ago, when I was still posting handwritten solutions to the homeworks in my graduate classes, I found some students submitting almost perfectly $\LaTeX$-formatted solutions. These solutions often contain far more details or theorems than needed, as one often only have to invoke a certain result covered in lectures, which I said “can be applied directly since we proved it”. Coincidentally, that was when Language models begin to come in more and more handy interfaces, I realized: maybe this is the time to ditch Homework as a way to assess the learning outcomes.

Some ideas

In the midterm of the Numerical PDE class this semester, I include the following optional semi-open-ended problems

Problem 1

Consider a higher-order approximation using a 5-pt stencil for the 1D Laplacian:

\[\Delta_{5, h} u_i :=\frac{1}{12 h^2}\left(-u_{i-2}+16 u_{i-1}-30 u_i+16 u_{i+1}-u_{i+2}\right)\]

whose truncation error is $|\tau| = \mathcal{O}(h^4)$. When using this stencil to approximate

\[\begin{cases}-u''=f \text{ in }(0,1),\\ u(0)=u(1)=0.\end{cases}\]

if one repeats the error analysis shown in the lecture (truncation error done, needs to estimate the eigenvalue of $A$ to bound $|A^{-1}|$), what \textbf{mathematical difficulty} can you foresee? e.g., is the $p$-th eigenvector still has this component form $\boldsymbol{v}_j = [\sin(p\pi j h)]$ for $j=1,\dots, n$? how about boundary treatment? You only need to share your thoughts on the difficulty together with light mathematical derivation on why it is difficult, no need to resolve or prove.

Problem 2

To approximate

\[\begin{cases}-u''=f \text{ in }(0,1),\\ u(0)=u(1)=0.\end{cases}\]

In lecture, we studied the continuous piecewise linear ($P_1$) finite element method on a uniform mesh. The resulting stiffness matrix has a tridiagonal structure closely related to the standard 3-pt stencil of the central finite difference discretization.

Now replace $P_1$ elements by continuous piecewise quadratic ($P_2$) elements on the same uniform mesh, which means that the approximation polynomial, restricted on each subinterval is now a quadratic polynomial. What mathematical difficulties can you foresee if we attempt to repeat the stiffness error analysis used for $P_1$ elements? You may choose at least 2 questions from the following list to discuss:

  • How many degrees of freedom does a $P_2$ element introduce, locally and globally, and how does this change the structure of the assembled stiffness matrix?
  • Are the eigenvectors of the stiffness matrix still $\boldsymbol v^{(p)}_j=\sin(p\pi jh)$?
  • Would the smallest eigenvalue of the stiffness matrix necessarily have the same scaling as that of the $P_1$ stiffness matrix?
  • How the quadrature formulas would change?

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