Hello, I want to start by saying how immensely helpful your videos have been to me , as I am studying Numerical Calculus within my major this year. The way you explain the concepts ends up roping me in this domain and I find myself wanting to know some more. That said, I do have a few questions. When do we choose whether our basis is linear or quadratic? Is it tied to how many nodes we choose to solve the ODE for? Additionally, what role do Chebyshev, Legendre, or Fourier bases play? Would you say they are the next step to take after I am done understanding and computing the approximations using these basis functions? Are they better at approximating than simple polynomials? Thank you once again for the amazing videos and explanations!
Thanks for the feedback, glad it could help. These videos are created for a introductory course on the finite element method for mechanical engineering students, so the background is mechanics. We mainly use low order polynomials because is cheap to integrate (using one or two function evaluations). For greater accuracy the domain is instead split into more subdomains (finite elements) and the approximation thus becomes piecewise. Yes, for one element, the number of nodes is the number of basis functions. The choice between linear or quadratic, for mechanics, comes into play when dealing with three dimensions, and PDEs. Usually the field variables of question need to be resolved and that can be done by either having more elements or a higher order approximation. You can find out more and get some context by visiting the course site: basicfem.ju.se/GalerkinPiecewiseSystematic/ The type of basis functions to use depends on the type of underlying function that you want to approximate. This method has similarities with Fourier which is used to approximate continuous, unbounded trigonometric functions.
@@mirzacenanovicThank you so much for the reply! Our professor glossed over most of the explanations regarding approximation methods, especially the Finite Methods and Spectral Methods, so most of these things are a bit confusing to me. I will definitely check out the lectures on the website! In addition to the video, the online Galerkin lecture was fantastic and it cleared out some of questions! If I may ask, could you recommend me some books/materials to study for numerical analysis?
