can you please tell me how the f(-0.75,-0.75) comes?

I am glad that you liked the video :) Unfortunately I won't make another video on finite differences, since I am not using this method. Instead I am using the Finite Element Method (FEM; see my other video) on a daily basis. In my opinion to see results quickly the Finite Difference Method is convenient, but to deal with complicated boundary conditions FEM is more straightforward. When it comes to solving the linear equation system you have two classes of solvers: direct and iterative solvers. For sparse matrices (from 2D problems) it is sufficient to use direct solvers, which are e.g. based on the LU decomposition, when you have less than 1 million unknowns. If you have more equations you might need to use iterative solvers with appropriate preconditioners, since direct solvers don't work anymore. As a first step I would advise you to try using a direct solver and only if that is infeasible to then use iterative solvers. But if you have an iterative solver that works this is obviously fine as well.

@@numericalanalysisbyjulianr4242 indeed, fair enough. Sad that it seems like FDM's use is simply as an introductory bridging tool for FEM since even FEM can be applied for 1-D problems. I just feel like it's kinda elegant the algebraic equations spring up from the differential operators. For 2D onwards, especially 3D and with complicated boundaries and heterogeneous material, FEM is the way to go. I've been intending on learning FEM too and demystifying this seemingly daunting topic. FEM is usually not taught in undergrad courses, but is often mentioned enough that one gets the importance and utility of it.

There is for example the area of optimal control of partial differential equations, where you try to minimize some goal functional J(y,u) such that that y solves a PDE that also involves the control u. If you want to learn more about this, check out the book "Optimal Control of Partial Differential Equations: Theory, Methods, and Applications" by Fredi Tröltzsch

You can check out my online explanation of multigrid for the Finite Element Method at julianroth.org/documentation/multigrid/index.html. The code for this is also on GitHub

Yes, I did my master in 1 year, since I wanted to get my current PhD position. This was certainly very stressfull, but I managed to do it somehow and now it is worth it :)

@@numericalanalysisbyjulianr4242 I guess you still have excellent grades 😅 I study at TU Berlin and can't even imagine taking 60 ECTS per semester. Maybe 45 would be possible, but 60 is incredible.

For time-dependent problems you can take a look at chapter 12 and especially section 12.3.2 in the lecture notes doi.org/10.15488/11709

When learning about Finite Elements or Finite Differences, you should at least have a basic understanding of calculus and linear algebra. You should know e.g. how to compute derivatives and integrals or how to work with vectors and matrices. To learn these topics, I recommend the playlists from 3blue1brown on these topics: www.3blue1brown.com/#lessons

That depends on the problem that you are solving. For example, if you have a uniform gravitational force pushing down you could have something like f(x, y) = -1 and thus f(-0.75, -0.75) = -1.

Thank you so much got it ❤

3blue1brown created a Python based animation library called manim (github.com/3b1b/manim), which he uses to create his videos. More recently, there has been a fork of manim, called manim community (www.manim.community/), which introduces some additional new features and quite significantly diverged from the original library from 3blue1brown. I used the community version of manim to create my videos.

@@numericalanalysisbyjulianr4242 sir please answer my question?