### HSL_MP62 Symmetric ﬁnite-element system: multiple-front method

The module HSL_MP62 uses the multiple front method to solve
sets of symmetric positive-deﬁnite ﬁnite-element equations
$\mathbf{AX}=\mathbf{B}$ that
have been divided into non-overlapping subdomains. The HSL routines MA62 and
MA72 are used with MPI for message passing.

TThe coeﬃcient matrix $\mathbf{A}$
must be of the form

$$\mathbf{A}=\sum _{k=1}^{m}{\mathbf{A}}^{\left(k\right)}$$

where the summation is over ﬁnite elements. The element matrix
${\mathbf{A}}^{\left(k\right)}$ is
nonzero only in those rows and columns which correspond to variables in the
$k$-th element. The
right-hand side(s) $\mathbf{B}$
may optionally be in the form

$$\mathbf{B}=\sum _{k=1}^{m}{\mathbf{B}}^{\left(k\right)}$$
where ${\mathbf{B}}^{\left(k\right)}$ is
nonzero only in those rows which correspond to variables in element
$k$.

In the multiple front method, a frontal decomposition is
performed on each subdomain separately. Thus, on each subdomain,
$L$ and
$U$ factors
are computed. Once all possible eliminations have performed within a subdomain, there
remain the interface variables, which are shared by more than one subdomain. If
${\mathbf{F}}_{i}$ is the remaining frontal
matrix for subdomain $i$,
and ${\mathbf{C}}_{i}$
is the corresponding right-hand side matrix, then the remaining problem
is

$$\mathbf{FY}=\mathbf{C},$$ | (1) |

where $\mathbf{F}={\sum}_{i}{\mathbf{F}}_{i}$ and
$\mathbf{C}={\sum}_{i}{\mathbf{C}}_{i}$. By treating
each ${\mathbf{F}}_{i}$
as an element matrix, the interface problem (1) is also solved by the frontal method.
Once (1) has been solved, back-substitution on the subdomains completes the
solution.

The element data and/or the matrix factors are optionally held in direct-access
ﬁles.