2,957 research outputs found
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
A semi-explicit multi-step method for solving incompressible navier-stokes equations
The fractional step method is a technique that results in a computationally-efficient implementation of Navier–Stokes solvers. In the finite element-based models, it is often applied in conjunction with implicit time integration schemes. On the other hand, in the framework of finite difference and finite volume methods, the fractional step method had been successfully applied to obtain predictor-corrector semi-explicit methods. In the present work, we derive a scheme based on using the fractional step technique in conjunction with explicit multi-step time integration within the framework of Galerkin-type stabilized finite element methods. We show that under certain assumptions, a Runge–Kutta scheme equipped with the fractional step leads to an efficient semi-explicit method, where the pressure Poisson equation is solved only once per time step. Thus, the computational cost of the implicit step of the scheme is minimized. The numerical example solved validates the resulting scheme and provides the insights regarding its accuracy and computational efficiency.Peer ReviewedPostprint (published version
A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations
In this paper a hybridized weak Galerkin (HWG) finite element method for
solving the Stokes equations in the primary velocity-pressure formulation is
introduced. The WG method uses weak functions and their weak derivatives which
are defined as distributions. Weak functions and weak derivatives can be
approximated by piecewise polynomials with various degrees. Different
combination of polynomial spaces leads to different WG finite element methods,
which makes WG methods highly flexible and efficient in practical computation.
A Lagrange multiplier is introduced to provide a numerical approximation for
certain derivatives of the exact solution. With this new feature, HWG method
can be used to deal with jumps of the functions and their flux easily. Optimal
order error estimate are established for the corresponding HWG finite element
approximations for both {\color{black}primal variables} and the Lagrange
multiplier. A Schur complement formulation of the HWG method is derived for
implementation purpose. The validity of the theoretical results is demonstrated
in numerical tests.Comment: 19 pages, 4 tables,it has been accepted for publication in SCIENCE
CHINA Mathematics. arXiv admin note: substantial text overlap with
arXiv:1402.1157, arXiv:1302.2707 by other author
Least-squares finite element method for fluid dynamics
An overview is given of new developments of the least squares finite element method (LSFEM) in fluid dynamics. Special emphasis is placed on the universality of LSFEM; the symmetry and positiveness of the algebraic systems obtained from LSFEM; the accommodation of LSFEM to equal order interpolations for incompressible viscous flows; and the natural numerical dissipation of LSFEM for convective transport problems and high speed compressible flows. The performance of LSFEM is illustrated by numerical examples
Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows
We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable
Free-energy-dissipative schemes for the Oldroyd-B model
In this article, we analyze the stability of various numerical schemes for
differential models of viscoelastic fluids. More precisely, we consider the
prototypical Oldroyd-B model, for which a free energy dissipation holds, and we
show under which assumptions such a dissipation is also satisfied for the
numerical scheme. Among the numerical schemes we analyze, we consider some
discretizations based on the log-formulation of the Oldroyd-B system proposed
by Fattal and Kupferman, which have been reported to be numerically more stable
than discretizations of the usual formulation in some benchmark problems. Our
analysis gives some tracks to understand these numerical observations
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