MA62: Sparse symmetric finite-element system: out-of-core frontal method

To solve one or more sets of sparse symmetric linear unassembled finite-element equations, \(\mathbf{AX} = \mathbf{B}\), by the frontal method, optionally holding the matrix factor out-of-core in direct-access files. The package is primarily designed for positive-definite matrices since numerical pivoting is not performed. Use is made of high-level BLAS kernels. The coefficient matrix \(\mathbf{A}\) must of the form

\[\mathbf{A} = \sum_ {k=1} ^ m \mathbf{A} ^{(k)}\] with \(\mathbf{A} ^{(k)}\) nonzero only in those rows and columns that correspond to variables in the \(k\)-th element.

The frontal method is a variant of Gaussian elimination and involves the factorization

\[\mathbf{A} = \mathbf{PLD} (\mathbf{PL}) ^T,\]

where \(\mathbf{P}\) is a permutation matrix, \(\mathbf{D}\) is a diagonal matrix, and \(\mathbf{L}\) is a unit lower triangular matrix. The solution process is completed by performing the forward elimination

\[(\mathbf{PL})\mathbf{DY}\normalfont = \mathbf{B},\]

followed by the back substitution

\[(\mathbf{PL}) ^ T \mathbf{X} = \mathbf{Y}.\]

MA62 stores the reals of the factors and their indices separately. A principal feature of MA62 is that, by holding the factors out-of-core, large problems can be solved using a predetermined and relatively small amount of in-core memory. At an intermediate stage of the solution, \(l\) say, the ‘front’ contains those variables associated with one or more of \(\mathbf{A} ^{(k)}\), \(k = 1, 2,..., l\), which are also present in one or more of \(\mathbf{A} ^{(k)}\), \(k = l+1,..., m\). For efficiency, the user should order the \(\mathbf{A} ^{(k)}\) so that the number of variables in the front (the ‘front size’) is small. For example, a very rectangular grid should be ordered pagewise parallel to the short side of the rectangle. The elements may be preordered using the HSL routine MC63.