Significance of FunctionSpace on calculation outcome

variational formulation
1

Hi guys,

This is a question more on understanding rather than code. I am wondering about the mathematical significance of the choice of function spaces over with our function to solve, u and our test function, v, are defined.

For example, if I chose my problem to be spanned by a function space defined by first order Lagrange Polynomials, compared to using a Crouzeix-Raviart system, for example: would this significantly change my outcome, or these basis functions chosen such that they are easier to compute?

Thanks again

2

As seen on DefElement, the first order Lagrange space and the Crouzeix-Raviart element span the same space, but the degrees of freedom are defined very differently.

This results in different properties, see for instance chapter 3.3.2 of the FEniCS book, Page 95