 # Finite Element Method (FEM)

## 2D depth-integrated flow (10)

The last step in the deduction of the finite element form is substituting the expressions for the approximate solutions into the weak form. The approximate solutions for the depth h and the flow velocities (u,v) for our unsteady, two-dimensional, depth-integrated problem can be written as follows using a semi-discrete method:

 where: and are the approximation functions and uim and hk are the respective function values at the element vertices with the vertex count M or K respectively.

The program RMA2 produces a vertex count of M=8 and 6 und K=4 and 3, using square approximation function for the velocities and linear approximation functions for the depth. It allows triangles and rectangles as element shapes. If the approximate solutions from Eq. 3-71 are substituted into the weak form (s. Eq. 3-64 and 3-70) and the assumptions of the Galerkin method are made, i.e. weighting function equals approximation function (W= und Q= ) the following expressions are obtained:

Continuity:  