### Local FE approximation in Dirichlet DD methods

#### by Yi Zhang

After decomposing a domain, we can formulate approximations by FE or BE methods in local subdomains. In the case of Dirichlet domain decomposition method, the global unknown on the skeleton is primal, i.e. , while local unknowns are the nodal variable within subdomain (FE) or on the local boundaries (BE). If focusing on a specific subdomain, one can neglect the subscripts and assume model problem as

By using Green’s identity, the weak formulation of this problem is

In which is the bilinear form for space . Assigning test function satisfying , we have

Now we decompose the primal into three components

which means, the unknown variable consists of its value within the domain, extension/prolongation from *unknown* Dirichlet boundary value on , and extension/prolongation from Dirichlet *data*. Here denotes the “extension/prolongation” operator, mapping a function as boundary value to a function on the domain. By this decomposition one has two unknowns, on the domain and on the boundary, so another weak formulation is needed. Assuming is the weak solution of governing equation, it is easy to show, for test function space

Finally, the weak formulation of the model problem is to resolve couple equations

for unknown and . Assigning two finite element spaces, one on , one on :

The Galerkin FE formulation of the coupled problem is then

Remembering that this formulation is for one subdomain, and in each subdomain same equations can be obtained as long as FEM is used to approximate interior solution. Take the summaion of those formulations across all the subdomains, and put them in matrix form:

Which is the well-known substructure method formulation in structure engineering. The subscripts imply “connecting” and “interior”. The unknown vectors are nodal solutions in the interior of subdomains and nodal solutions on the skeletons. Classical substructure algorithm goes like this:

- Find , then ;
- ;
- ;
- ;

Even though the inverse is easy to be parallelized due to the matrix being block diagonal, this algorithm is impractical for large problem since in step 4, finding schur complement is costly. Preconditioner are always used instead of solving the matrix equation in step 4.