The Schur Complement Method of Inverting Global Operators

We are left with a matrix inversion problem. The matrix ${\it A} = Z^t L Z$ has to be inverted. The Z matrix groups together the degrees of freedom for the boundary mode shapes for all the elements. The remaining interior degrees of freedom are collected in one group per element. The matrix now has the form shown in figure 4.18.

  

Figure 4.18: Form of the global operator matrix. Notice the sparsity of A and the block nature of B and C.

We can take advantage of the bandedness of the A boundary-boundary matrix, and the decoupled C matrix. The system can be rewritten as:

$$\left[\begin{array} {cc} A-B C^{-1} B^T & 0 \ B^T & C \end... ...begin{array} {c} f_b - B C^{-1} f_i \ f_i \end{array}\right]$$

where $\phi_b$ are an array of the unknown global boundary coefficients and $\phi_i$ are the unknown local interior coefficients.