19th March 2020

# 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

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

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

$\mathbf{B} = \sum_{k=1}^ m \mathbf{B} ^{(k)}$ where $$\mathbf{B} ^{(k)}$$ 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 $$\mathbf{F} _{i}$$ is the remaining frontal matrix for subdomain $$i$$, and $$\mathbf{C} _{i}$$ is the corresponding right-hand side matrix, then the remaining problem is

$\label{mp43.eqn} \mathbf{FY} = \mathbf{C},$ 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 ([mp43.eqn]) is also solved by the frontal method. Once ([mp43.eqn]) 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.