Variational based numerical solution of PDEs

by Yi Zhang

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