5.6.1 The Schur complement

The Schur complement is named in honor of the mathematician Isaai Schur^5.1 (1894-1939 working in Germany). We introduce it from a FE point of view.

Figure 5.5: 2 super elements (=subdomains) with discretization

The distribution of the FE-node in Fig. 5.5 implies a block structure in the system of equations (5.1) :

$$\begin{pmatrix}K_C & K_{CI,1} & K_{CI,2} \ K_{IC,1} & K_{I,1} &... ...e{f}_C \ \underline{f}_{I,1} \ \underline{f}_{I,2} \end{pmatrix} \enspace .$$ (5.6)

A Gauß-elimination of the upper right entry of the matrix results in the cascaded system of equations

$$\begin{pmatrix}S_C & 0 \ K_{IC} & K_I \end{pmatrix} \cdot \begi... ...;\; \begin{pmatrix}\underline{g}_C \ \underline{f}_I \end{pmatrix} \enspace ,$$ (5.7)

with the Schur complement

$$S_C = K_C - K_{CI} K_I^{-1} K_{IC}$$ (5.8)

$$= \left( K_{C,1} - K_{CI,1} K_{I,1}^{-1} K_{IC,1} \right) + \left( K_{C,2} - K_{CI,2} K_{I,2}^{-1} K_{IC,2} \right)$$

$${\text{and a modified right hand side}}$$

$$\underline{g}_C = \underline{f}_C - K_{CI} K_I^{-1} \underline{f}_I \enspace .$$

Therein, $K_{IC}= \left(\begin{smallmatrix}K_{IC,1} \ K_{IC,2} \end{smallmatrix}\right)$ and $K_I = \textrm{blockdiag}\{K_{I,1},K_{I,2}\}\enspace$ are block matrices. The matrix $S_C$ is in general fully occupied - the Schur complement can be given explicitly in special cases.

The main advantage of system (5.7) consists in the fact that due to the block structure of $K_I$ the inversion of the local matrices $K_{I,1}$ and $K_{I,2}$ can be done successively on each super element. Therefore it possible by means of storing unused data on disc to solve problems which would otherwise exceed the main memory. This was used intensively in the 60/70-ies by engineers and is well-known as Element-by-Element method (EBE).

Because of the expensive inversion of the Schur complement (no more a sparse matrix) the system in II) will be often solved by means of an iteration method. The great advantage is that we need therein only matrix-times-vector multiplications with the Schur complement on the super elements :

$$\underline{r}_C := S_C \underline{w}_C$$

$$= \sum\limits_{s=1}^P \left( K_{C,s} - K_{CI,s} K_{I,s}^{-1} K_{IC,s} \right) \cdot\underline{w}_{C,s} \enspace .$$

The inversion of matrices $K_{I,s}$ will be done in each super element by a LU- or Cholesky-factorization so that in the following the application of $K_{I,s}$ consists only in the elimination steps.