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- I know how to derive the 1D finite element formulation using the Galerkin linear weighting functions (in the end I studied this myself as the classes I took did not teach this).
- I know that you can mechanically solve the 1D formulation using the generic force = stiffness matrix × displacement method (this is what I initially knew before taking the undergrad and graduate classes, and yet this is the one they teach here. Yep, waste of money).
- I also know the shape functions for 2D (the triangle thing, where T(x,y) = T
_{i}(x_{i},y_{i})N_{i}(x,y) + T_{j}(x_{j},y_{j})N_{j}(x,y) + T_{k}(x_{k},y_{k})N_{k}(x,y) then the weighting functions are N_{i}(x,y), N_{j}(x,y), and N_{k}(x,y), which is integrated to the differential equations as in 1D).

I actually wanted to avoid those things as I don't know how they would be relevant to me as I'm not really a mathematics major/graduate, I'm an engineering graduate. I am truly sorry for asking help here if it's too much to ask as I really wanted to know where did these things come from, or at least guide me until I achieve the 'unsimplified' but 'clear' solution I wanted - what I'm hoping for is someone be able to guide me with the pertinent theorems in order to arrive with the default, unsimplified formulation similar to the 1D case; something like:

T

_{i}× (insert lengthy integral) + T

_{j}× (insert lengthy integral) + T

_{k}× (insert lengthy integral) = 0.

Anyway, I started off by using the Laplace equation. This is what I have now:

Anyway, these are the questions.

- I know I am supposed to multiply the entire PDE with the weighting function. Which weighting function do I use? I remember in the 1D case you have to integrate the differential equations twice, one for the i
^{th}case and one for the j^{th}case. Am I correct to assume that I integrate this differential equation thrice, seeing there are three vertices in the triangular element? So that would mean I integrate the PDE; one for the i^{th}, one for the j^{th}case, and one for the k^{th}case? - I am assuming that you have to use double integration along the x-axis and the y-axis. My problem is about the limits. Which limits do I use? If the bottom limit of the integrands are x
_{i}and y_{i}, what would be the upper limit?