I'm quite a newbie to numerical simulation (heat transfer) and I'm quite confused about a sentence that our teacher said. He said
"Finite Difference Method (FDM) and Boundary Element Method (BEM) are more accurate than Finite Element Method (FEM). Moreover, in FDM and BEM, the energy is always conserved. However in FEM, it's not always."
He didn't explain why and then disappeared. Maybe I misheard or missed something in his class. Based on his statement, I have following questions.
Is his statement really true?
I understand that both FEM and BEM start from the weak form of the PDE, e.g. \begin{equation} \ \int_{\Omega} w(-\nabla \cdot (-\lambda \nabla T)) d\Omega =0. \end{equation} By definition, the test function
win the weak form should be arbitrary. However, in both FEM and BEM, some specific test functions are selected (e.g. Galerkin's method in FEM and Green's function in BEM). Does this mean that the equation above only holds with the chosen test function? If I use other test functions to test the accuracy, will the weak form equation not hold? Does this mean that the energy is not conserved in both methods?
It would be also appreciated if anyone could recommend some literatures or books addressing this question.