Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

This paper studies fully discrete approximations to the evolutionary Navier{ Stokes equations by means of inf-sup stable H1-conforming mixed nite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound O(hk) for the L2 error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.MINECO grant MTM2016-78995-P (AEI)Junta de Castilla y León grant VA024P17Junta de Castilla y León grant VA105G18MINECO grant MTM2015-65608-

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier-Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad-div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at $Re_{\tau}=180$ as well as $590$.Comment: 28 pages, in preparation for submission to Journal of Computational Physic

A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology

Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

This is a post-peer-review, pre-copyedit version of an article published in Journal of Scientific Computing. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10915-019-00980-9This paper studies fully discrete approximations to the evolutionary Navier–Stokes equations by means of inf-sup stable H1-conforming mixed finite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound O(hk) for the L2 error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this methodInstituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Spain. Research supported under grants MTM2016-78995-P (AEI/MINECO, ES) and VA024P17, VA105G18 (Junta de Castilla y León, ES) cofinanced by FEDER funds ([email protected]) Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain. Research supported by Spanish MINECO under grant MTM2015-65608-P ([email protected]) Departamento de Matemáticas, Universidad Autónoma de Madrid. Spain Research supported under grants MTM2016-78995-P (AEI/MINECO, ES) and VA024P17 (Junta de Castilla y León, ES) co financed by FEDER funds ([email protected]