9,179 research outputs found
A Bramble-Pasciak conjugate gradient method for discrete Stokes equations with random viscosity
We study the iterative solution of linear systems of equations arising from
stochastic Galerkin finite element discretizations of saddle point problems. We
focus on the Stokes model with random data parametrized by uniformly
distributed random variables and discuss well-posedness of the variational
formulations. We introduce a Bramble-Pasciak conjugate gradient method as a
linear solver. It builds on a non-standard inner product associated with a
block triangular preconditioner. The block triangular structure enables more
sophisticated preconditioners than the block diagonal structure usually applied
in MINRES methods. We show how the existence requirements of a conjugate
gradient method can be met in our setting. We analyze the performance of the
solvers depending on relevant physical and numerical parameters by means of
eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of
the relevant preconditioned sub-matrices. We illustrate our findings using the
flow in a driven cavity as a numerical test case, where the viscosity is given
by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a
Bramble-Pasciak conjugate gradient method with block triangular preconditioner
outperforms a MINRES method with block diagonal preconditioner in terms of
iteration numbers.Comment: 19 pages, 1 figure, submitted to SIAM JU
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
In this work, we apply stochastic collocation methods with radial kernel
basis functions for an uncertainty quantification of the random incompressible
two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the
existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase
Navier-Stokes equation for each given realization. We are able to empirically
show that the resulting kernel-based stochastic collocation is highly
competitive in this setting and even outperforms some other standard methods
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
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