Taking the Robin boundary conditions into account

 

Computation of the right-hand side :

The vector which must be assembled is defined by :

with :

- i are shape functions defined on the reference interval
- is the number of nodes per edge
- n1 and n2 are the start and end nodes of the edge
- are uA are the values of the boundary condition on the edge

 


 

Computation of the element matrix :

The matrix which must be assembled is defined by :

 


 

Taking the boundary conditions of first type into account :

To take these boundary conditions into acount, we modify the right-hand size using the following method :

with :

- g1(xj) : the value of the Dirichlet boundary condition at the point xj
- : containing the numbers of the nodes of the edge E(e3) which do not have Dirichlet boundary conditions.
- : containing the numbers of the nodes of the edge E(e3) which have Dirichlet boundary conditions.

 

We also modify the stiffness matrix with :

with : containing the numbers of the nodes of the edge E(e3)

 


 

Assembling process :

For each border domain of type 3

For each edge of the domain

Computation of f(e3) und K(e3)

Modification of f(e3) and K(e3) to take the boundary conditions of first type into account

For each node of the edge, which has i as global number and k as local number

For each node of the edge, which has j as global number and l as local number