DISCRETIZATION WITH THE X-FEM

We now describe how to discretize this BVP with the X-FEM.The standard finite element approximation to the temperature field takes the form

where are the classical nodal shape functions and are the corresponding nodal coefficients. The sum is taken over all nodes in the mesh.

In the X-FEM, we extend the above using the construction

where is the enrichment function, are enriched degrees of freedom and is a subset of the nodes in the mesh. For example, we may take to consist of all nodes whose shape function supports contain the location of the zone of local behavior :

This approach allows the approximation to follow a moving feature of interest, as both the set and the function can evolve with . However we note that this approach also requires a-prior knowledge of the motion of the local zone .

In the case of the heat equation (), the mass and stiffness matrices are given by:

where are the weight functions and can be the classical shape functions or the function .