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

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 $AX = B$ that have been divided into non-overlapping subdomains. The HSL routines MA62 and MA72 are used with MPI for message passing.

TThe coeﬃcient matrix $A$ must be of the form

A = \sum_{k = 1}^{m} A^{(k)}

where the summation is over ﬁnite elements. The element matrix $A^{(k)}$ 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^{(k)}

where $B^{(k)}$ 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 $F_{i}$ 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 (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.