 # Finite Element Method (FEM)

## The weak form (5)

In order to transfer the weak form into a finite element form in the next step, the finite element approximation for T  from Eq. 3-11 is substituted into Eq. 3-16 and we obtain:

After a few modifications, this equation can be written in matrix form

The matrix [Ke] is the so-called coefficient matrix, also called stiffness matrix in statics. Eq. 3-19 above is quasi a first important milestone in our efforts to understand the finite element method, that is to say the (weak) finite element form of Eq. 3-7. In order to derive the weak form of the depth-averaged shallow water equation in chapter 3.1.8, we could actually make a break at this point, but unfortunately one who has the inconvenience of having to make practical use of the finite element method is confronted with a number of mathematical-numerical problems.

The approaches for solving these problems have an influence on the solution of course. When evaluating the result, the user (for example of a commercial software) not only has to be able to judge the preconditions and the validity of the basic mathematical equation, but also the mathematical-numerical methods that are used. For this reason, additional questions will be discussed at this point like element connections and setting up the global system of equations, properties of and requirements for the approximation functions, numerical integration and time-discretization for unsteady problems.  