The Galerkin operator

The basis functions I use are bilinear functions. The nodal basis function for the i^th node N_i has the value 1 at node i, zero at all other nodes, and is a bilinear function in between. Figure  shows the basis function for one node. The element A_i,j of the Galerkin operator is the integral of the dot product of the gradient of the basis functions for the i^th and j^th nodes. The basis function for the i^th node is only non-zero in the four elements that have node i as a vertex. Thus, the only non-zero elements of the operator are A_i,j where nodes i and j are vertices of a common element.

 

nodaldiag
Figure 2 The basis function for a node is one at the node and only non-zero in the four elements connected to the node.

Almost all FE codes try to avoid constructing the full Galerkin operator because it is a very sparse matrix. Instead they construct local operators for each element. In the model problem the element operator is a $4\times 4$matrix, where the elements are the evaluation of the integral a(N_i,N_j) over the element. Here, i and j run from 1 to 4, the local node numbering for the element. The global operator is formed as a sum of the element matrices. The application of the operator can be performed as a three stage operation,

1.

spread the nodal values to form a 4-vector for each element,

2.

parallel application of the element matrices,

3.

gather the RHS 4-vector values and sum into the output vector.

This is the standard way that unassembled Galerkin operators are applied on vector computers (Ferencz, 1989). These calculations are performed for each element so the vector length is the total number of elements.