Galerkin Finite Element Method

With this method, the spatial variation of the state and flux variables are defined in terms of the nodal values of the quantities and a local interpolation function.[^1] If the input is $U$ and the state vector is $x$ then the discretization is
$U(x)={\sum }_i{N}_i(x)U_i(t)$
Where $N_i$ is the interpolation function that has a value of 1 at the node $i$ and 0 elsewhere.[^1] The equations are projected onto a test function and integrated over a domain.[^1]
$M_{ij}\frac{d{U}_i}{dt}=-R_{x,ij}{F}_i-R_{y,ij}{G}_i-R_{z,ij}{H}_i$
Where $M$ is the mass matrix, and the $R$ matrices are the residual matrices defined by integration over the entire domain.[^1]

Stabilized Galerkin Finite Element

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