Finite Element Methods on a Time-Varying System

where Ω ⊂ R, ∂ΩD ∪ ∂ΩN = ∂Ω and the two boundary regions are distinct with ∂ΩD non-empty. Specifically, we will consider the key results in the development of the theory of finite elements, including weak solutions and the structure of the individual elements, as well as the numerical implementation of the method. We will subsequently implement a finite element numerical solver in order to solve this equation in a region of space, with f , gD and gN prescribed. We will then extend these concepts in order to consider a time-varying problem using Euler time-steps, and finally implement a finite element solution to a simple time-varying analogue of the original spatial partial differential equation, given as