### Variational based numerical solution of PDEs

#### by Yi Zhang

For problem , the variational type methods are based on the weak form

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

1. Bubnov-Galerkin: test function space

- Finite element method: consists of splines with local support. Conforming FEM: , together with same bilinear form and linear form in discrete variational equation. Non-conforming FEM: or (similar for ) or both, which results in discrete variational eqution

- Spectral method: consists of orthogonal polynomials.

2. Petrov-Galerkin:

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