I'm a PhD working in a computational mechanics lab. I come from a Math department, and I have a good background for what concerns the basics of finite elements, like inf-sup conditions, DG, non-conforming FEs. I also did some basic codes, like Poisson on a square/L-shape/etc. (or other elliptic problems) using the classical building blocks of a code, like local to global mapping, quadrature formulas on the reference, etc.
However, there's something that I really don't know how to do, and this is how to solve vector valued problems. I know how to solve them in deal.II, for instance, and I watched the related video-lecture about it. But I do really need to re-invent the wheel (in 2D) this time. Let me try to explain: when we're using our software(s) to solve scalar problems, I really know what is going on inside the assembly routine and I know I would be able to replicate it (of course not in terms of efficiency), but that's not the case for the linear elasticity equation, for instance. I know that the test functions are vector valued, but I'm lacking the ability to put this into code. In particular, I'm interested in the approach where each vector basis function has only one non-zero component.
So I am looking for some reference (lecture notes, books, whatever) where I can find a simple and didactic explanation about the way the code has to be organised for this equation and the relative code. If there's something with MatLab that would be perfect, as I only need to understand the basic building blocks. However, I can read with no problems also Python and C++ codes.