In this presentation we examine the convergence characteristics of a
Krylov subspace solver preconditioned by a new indefinite
constraint-type preconditioner, when applied to discrete systems
arising from low-order mixed finite element approximation of the
classical biharmonic problem. The preconditioning operator leads to
preconditioned systems having an eigenvalue distribution consisting of
a tightly clustered set together with a small number of outliers. We
compare the convergence characteristics of a new approach with the
convergence characteristics of a standard block-diagonal Schur
complement preconditioner that has proved to be extremely effective in
the context of mixed approximation methods.
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In the second part of the presentation we are concerned with the
efficient parallel implementation of proposed algorithm on modern
shared memory architectures. We consider use of the efficient parallel
"black-box'' solvers for the Dirichlet Laplacian problems based on
sparse Cholesky factorisation and multigrid, and for this purpose we
use publicly available codes from the HSL library and MGNet collection.
We compare the performance of our algorithm with sparse direct solvers
from the HSL library and discuss some implementation related issues.