Version 2.1.0

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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.