2,926 research outputs found
The cost of continuity: performance of iterative solvers on isogeometric finite elements
In this paper we study how the use of a more continuous set of basis
functions affects the cost of solving systems of linear equations resulting
from a discretized Galerkin weak form. Specifically, we compare performance of
linear solvers when discretizing using B-splines, which span traditional
finite element spaces, and B-splines, which represent maximum
continuity. We provide theoretical estimates for the increase in cost of the
matrix-vector product as well as for the construction and application of
black-box preconditioners. We accompany these estimates with numerical results
and study their sensitivity to various grid parameters such as element size
and polynomial order of approximation . Finally, we present timing results
for a range of preconditioning options for the Laplace problem. We conclude
that the matrix-vector product operation is at most \slfrac{33p^2}{8} times
more expensive for the more continuous space, although for moderately low ,
this number is significantly reduced. Moreover, if static condensation is not
employed, this number further reduces to at most a value of 8, even for high
. Preconditioning options can be up to times more expensive to setup,
although this difference significantly decreases for some popular
preconditioners such as Incomplete LU factorization
A Combined Preconditioning Strategy for Nonsymmetric Systems
We present and analyze a class of nonsymmetric preconditioners within a
normal (weighted least-squares) matrix form for use in GMRES to solve
nonsymmetric matrix problems that typically arise in finite element
discretizations. An example of the additive Schwarz method applied to
nonsymmetric but definite matrices is presented for which the abstract
assumptions are verified. A variable preconditioner, combining the original
nonsymmetric one and a weighted least-squares version of it, is shown to be
convergent and provides a viable strategy for using nonsymmetric
preconditioners in practice. Numerical results are included to assess the
theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure
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