## Version 2.0.0

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### 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

 $FY=C,$ (1)

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.