### HSL_MP42 Unsymmetric ﬁnite-element system: multiple-front method, element
entry

The module HSL_MP42 uses the multiple front method to solve sets of ﬁnite-element
equations $AX=B$
that have been divided into non-overlapping subdomains. The HSL routines MA42 and
MA52 are used with MPI for message passing.

The coeﬃcient matrix $A$
must be of the form

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

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

$$B=\sum _{k=1}^{m}{B}^{\left(k\right)}$$
where ${B}^{\left(k\right)}$ 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 together with any variables that
are not eliminated because of stability or eﬃciency considerations. If
${F}_{i}$ is the remaining frontal
matrix for subdomain $i$,
and ${C}_{i}$
is the corresponding right-hand side matrix, then the remaining problem
is

where $F={\sum}_{i}{F}_{i}$ and
$C={\sum}_{i}{C}_{i}$. By treating
each ${F}_{i}$
as an element matrix, the interface problem (3) 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.