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

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

This is post-peer-review, pre-copyedit version of an article published in Journal of Scientific Computing. The final of authenticated version is available online at: https://doi.org/10.1007/s10915-017-0447-2A two-grid scheme to approximate the evolutionary Navier–Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size H at any positive time T>0 . Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size h<H . The method increases the rate of convergence of the standard Galerkin method in one unit in terms of H and equals the rate of convergence of the standard Galerkin method over the fine mesh h. However, the computational cost is essentially the cost of approaching the Navier–Stokes equations with the plain Galerkin method over the coarse mesh of size H since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval (0, T]. For the analysis we take into account the loss of regularity at initial time of the solution of the Navier–Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shownJ. Novo: Research supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

Vector potential methods

Vector potential and related methods, for the simulation of both inviscid and viscous flows over aerodynamic configurations, are briefly reviewed. The advantages and disadvantages of several formulations are discussed and alternate strategies are recommended. Scalar potential, modified potential, alternate formulations of Euler equations, least-squares formulation, variational principles, iterative techniques and related methods, and viscous flow simulation are discussed

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Cumulative reports and publications thru 31 December 1982

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