### Three-Body Problem Series

I finished this series relatively quickly, probably in the span of a month total for three books. Looking back, the best book was probably the first two. There was just an air of mystery surrounding the nature of the invading aliens. Who are they? Why are they coming? What kinds of technology do they have? These questions really drives the first novel into a satisfying conclusion.

In the second and third books, where time skips anywhere from one to a few dozen years, a bleak picture of the universe is painted by Liu (the author). To no surprise, the universe of the novel is populated with lifeforms who mistrust each other and seek to destroy one another. Everything is explained quite thoroughly, but sometimes a bit too much. I wish he left some deduction for the readers to make ourselves rather than spoon feeding all the details.

There are a few more criticisms I have of the second and third books:

1. The character development falls mostly flat. I really didn’t care about any of them but rather the state of humanity as a whole. In contrast, the first book contained a fascinating historical overview set in communist China which helped build the characters.
2. Liu is really quite imaginative in the types of weapons that an alien with far superior technology can employ.  Unfortunately, some of them seem quite farfetched. I just think if they posses the power to alter reality, there would be better ways of waging war.
3. Multiple times, Liu thinks that society as a whole would “agree” on an idea. As we see in our current political situation, this really doesn’t make any sense.

### Mixed Finite Element Formulation and the necessity $H(\textrm{div})$

One can solve Poisson’s problem $-\Delta u = f$ in $d$ dimensions with homogeneous Dirichlet boundary conditions using a mixed formulation as explained below:

Let $\sigma = \nabla u$, then for a sufficiently smooth function $\tau$, by Green’s theorem
\begin{align*}
(\sigma, \tau) &= (\nabla u, \tau) \\
&= -(u, \textrm{div } \tau).
\end{align*}
Again, choosing $v$ a function sufficiently smooth, we have
\begin{align*}
f = -\textrm{div } \sigma \implies (f, v) = (-\textrm{div } \sigma, v).
\end{align*}
This gives the saddle-point problem: find $(u, \sigma) \in V \times M$ such that
\begin{align*}
(\sigma, \tau) + (u, \textrm{div } \tau) &= 0\\
(\textrm{div } \sigma, v) &= -(f, v)
\end{align*}
hold for all $(\tau, v) \in V \times M$. Note that we don’t have to take a derivative of $u$, hence it’s natural to try $M = L^2$, but what about the space $V$?

One very easy choice to guess is $V = [H^1(\Omega)]^d$ as we want the divergence to be all defined, but unfortunately this doesn’t work as the gradient of the solution to Poisson’s problem can easily not be in $[H^1(\Omega)]^d$.

In order to illustrate this, consider $u =\left(r^{2/3}-r^{5/3}\right) \sin \left(\frac{2 \theta }{3}\right)$ on the domain of the unit circle with bottom left quarter taken out. It’s not hard to see that $u = 0$ on the boundary of the domain, and we can easily find the $f$ such that it satisfies Poisson’s equation. Now, we can either calculate the gradient exactly or argue as follows.

First, recall how to take a gradient in polar coordinates. Note that $\partial_r u \approx r^{-1/3}$ plus higher order terms and that $\frac{1}{r}\partial_\theta u \approx r^{-1/3}$ plus higher order terms also. Now, one can easily calculate the $H^1$ seminorm to see that the derivative is unbounded as we’re integrating over $[0,1]$ with $(r^{-4/3})^2r$ terms (the extra $r$ comes from the change of basis from polar integration).

The above is an example of why the space $H(\textrm{div})$ is needed.

### Celeste

Once again, Dunkey has proven himself to be a modern day Donald Draper… in some sense. I bought Celeste almost strictly due to how fun it seemed. It truly is a great game with tight controls and extremely interesting level design.

The first point is starting to get quite standard now, but I want to reiterate on the latter point. It seems many high-ceiling platformers like Supermeatboy or the age-old N game constantly rely on precision. Celeste throws that away with an emphasis on when/where/how you use the air dash. That air dash, that one extra mechanic, really is the crux. Honestly, it reminded me of Ori’s dash, but the level design here is more like a puzzle.

Of course, the game can get a bit annoying. There are places where precise timing is the only way through the level (or so it seems to me?). Flag number 9 is particularly annoying. One can turn on the assist mode, but it really breaks the game by making it very easy. Another quirk is that it doesn’t save automatically when quitting from the Switch; this caused me to have to beat certain levels twice as I was switching between games.

### The Pattern is a Lie

We started a new problem recently, but still within the realms of preconditioning. The exact question and algorithms will remain under wraps right now, but the preliminary results I got were incredibly curious; here’s a plot of condition number below:

Note that a constant condition number as we increase order is extremely good, but my adviser and I were not rejoicing.  It’s just that the result was too good! We were not expecting the condition number to not increase, as that’s far better than what current literature (and our intuition) suggests.

After a week of fumbling around, I’ve extended the computation and produced the following plot:

Not that the above plot is the same algorithm, just with extended order (or p). And ahhh, there it is,  the growth that we were looking for.

Even though it’s a worse result that what we initially saw, we can actually prove this version!

### The Goldfinch

What a weird novel. It gradually crosses the line from a coming-of-age story (a Bildungsroman apparently) to a crime novel.

The first portion was beautifully done, with the main character characterized in-depth alongside his partner. The novel began with a tragedy befalling our young protagonist, and a quick introduction to all the relevant main characters. Chekhov’s gun really applied in this case, where a girl described within the first some 30 (?) pages becomes the overarching love interest.

From here, the character moves to Las Vegas and probably the best part of the book. There, the character really developed and really solidifies as a character. Another tragedy befalls our little teenager and he is forced to move back to NYC.

The final act of the book is really lackluster. The author introduces far too many characters and plot points which I didn’t understand (or care to for that matter). It was just too much and an abrupt change in pace. Overall, an incredibly lengthy novel which I just don’t think is that good…