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1.5 Correspondence: operator-module

The solution of a linear system consists of several steps which can be decomposed into several mathematical operators:

• construction of the matrix and right-hand-side,
• solution.

The construction of the matrix can be decomposed into:

• compute the pointers defining the matrix structure,
• matrix assembly,
• assemble the right-hand-side,
• impose boundary conditions.

depending on the method chosen, direct or iterative, the solution can also be decomposed into several operators:

• Direct methods:
  • matrix factorization,
  • forward-and-backward substitution.
• Iterative methods:
  • construction of the preconditioner,
  • iterative algorithm.

For the particular case of the domain decomposition method, all the above operators are used, as this algorithm performs iterations on the unknowns on the sub-domain interfaces, the unknowns inside the sub-domains being recomputed at each iteration by a direct method.

This sequence can be summarized as follows:

• For each sub-domain:
  • assembly and factorization of the matrix of the Dirichlet problem,
  • assembly and factorization of the matrix of the Neumann problem,
  • construction of the right-hand-side.
• For the global solution:
  • construction of the preconditioner,
  • conjugate gradient solution algorithm on the interface, consisting at each iteration of a Dirichlet forward-and-ackward substitution and a Neumann forward-and-backward substitution on each sub-domain.

As a function of the type of method chosen, on the one hand, and the matrix characteristics, on the other hand, there are several modules in library RESO corresponding to each operator:

• Computation of pointers :
  • PREPAC  for a MUA structure,
  • PREPAF  for a frontal method,
  • PREPGC  for a AMAT structure.

• Matrix assembly :
  • ASSAMA  for a AMAT structure,
  • ASMAPS  for a MUA structure in secondary memory,
  • ASSMUA  for a MUA structure in main memory.
• Assembly of the right-hand-side vector :
  • ASEMBV  for a B structure in main memory,
  • ASMBMS  for a B structure in secondary memory.
• Applying the boundary conditions :
  • CLIMGC  for a AMAT structure,
  • CLIMPC  for a MUA structure in main memory,
  • CLIMPS  for a MUA structure in secondary memory.
• Solution by a direct method:
  • Factorization :
    • CHOLPC  for positive definite symmetric matrices (structure MUA symmetric in main memory),
    • CHOLPS  for positive definite symmetric matrices (structure MUA symmetric in secondary memory),
    • CROUPC  for symmetric matrices (structure MUA symmetric in main memory),
    • GAUSPC  for regular matrices (structure MUA non-symmetric in main memory).
  • Solution by forward-and-backward substitution :
    • DRCHPC  for positive definite symmetric matrices (structure MUA symmetric in main memory),
    • DRCHPS  for positive definite symmetric matrices (structure MUA symmetric in secondary memory),
    • DRCRPC  for symmetric matrices (structure MUA symmetric in main memory),
    • DRGAPC  for regular matrices (structure MUA non-symmetric in main memory).
  • Special case of the frontal method :
    • FRONT : this model performs the tasks consisting of the assembly of the matrix and RHS, imposing the boundary conditions, the factorization of the matrix and, lastly, the solution of the system, whatever the properties of the matrix.
• Solution by an iterative method:
  • Construction of the preconditioner :
    • CONDLU  incomplete factorization of regular matrices (structure
      [4] AMAT non-symmetric in main memory),
    • FANIGC  incomplete factorization of positive definite symmetric matrices (structure AMAT symmetric in main memory).
  • Iterative algorithm :
    • DGRADA  iterations of double conjugate gradient  and preconditioned for regular matrices (structure AMAT in main memory)
    • ICHRGC  conjugate gradient iterations preconditioned by incomplete factorization  of Cholesky or Crout, for positive definite symmetric matrices (structure AMAT in main memory)
    • GCDIAG  conjugate gradient iterations with a diagonal preconditioning   for positive definite symmetric matrices (structure AMAT in main memory),
    • SSORGC  conjugate gradient iterations with a SSOR preconditioning , for positive definite symmetric matrices (structure AMAT in main memory),
    • SIMPGC  non preconditioned conjugate gradient iterations for positive definite symmetric matrices (structure AMAT in main memory).

For the case of a domain decomposition method, we need to use specific operators (super modules that use the preceding algorithms):

• PRSDOM : this program creates, using D.S. NOPO of different sub-domains as well as D.S. MILI and FORC, the operators required by the conjugate gradient algorithm,
• SDOMVR : this program contains the conjugate gradient algorithm for the Schur matrix complement  in single precision,
• SDOMVD : this program contains the conjugate gradient algorithm for the Schur matrix complement in double precision.