### Variational based numerical solution of PDEs

For problem $Lu=f$, the variational type methods are based on the weak form

$a(u,v)=(f,v),\quad \forall v\in V$

The numerical methods differs at the point of in which test function space is and in which function space $U$ the solution $u$ is searched.

1. Bubnov-Galerkin: test function space $V_h,\quad U=V_h$

• Finite element method: $V_h$ consists of splines with local support.  Conforming FEM: $V_h\subset V$, together with same bilinear form $a(\cdot,\cdot)$ and linear form $f(\cdot)$ in discrete variational equation. Non-conforming FEM: $V_h\nsubseteq V$ or $a_h(\cdot,\cdot)\neq a(\cdot,\cdot)$ (similar for $f$) or both, which results in discrete variational eqution $a_h(u_h,v_h)=f_h(v_h)$
• Spectral method: $V_h$ consists of orthogonal polynomials.

2. Petrov-Galerkin:$\quad U\neq V_h$