Coordinate transformation and numerical quadrature
In Galerkin method, the discretization of the governing equation is achieved through converting it into a weak formulation with a finite number of trial and test functions. The weak formulation consists of integrals defined in the simulation domain and Neumann/Robin boundaries. To calculate these integrals defined on physical domain, the general strategy, in one sentence, is to convert it into an equivalent integral define on a corresponding reference domain by coordinate transformation and integrate by numerical quadrature. To illustrate how it works, we first consider a one-dimensional case with both coordinate transformation and numerical quadrature, then extend it to higher dimensional cases. ...