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We study the qualitative behavior of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, on the unit square or cube subject to various types of boundary conditions. It is shown that for given initial data in H3(Ω), under the assumption that only the entropic energy associated with the initial data is small, there exist global-in-time classical solutions to the initial-boundary value problems of the model subject to the Neumann-Stress-free and Dirichlet-Stress-free type boundary conditions; these solutions converge to equilibrium states, determined from initial and/or boundary data, exponentially rapidly as time goes to infinity. In addition, it is shown that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model as the chemical diffusion rate tends to zero under the Neumann-Stress-free type boundary conditions. Numerical analysis is performed for a discretization of the model with the Dirichlet-Stress-free type boundary conditions, and a monotonic exponential decay to the equilibrium solution (analogous to the continuous case) is proven. Numerical simulations are supplemented to illustrate the exponential decay, test the assumptions of the exponential decay theorem, and to predict boundary layer formation under the Dirichlet-Stress-free type boundary conditions. [ABSTRACT FROM AUTHOR]

We prove that for several inf-sup stable mixed finite elements, the solution of the Chorin/Temam projection methods for Navier–Stokes equations equipped with grad–div stabilization with parameter γ converge to the associated coupled method solution with rate γ−1 as γ → ∞. We prove this result for both backward Euler schemes and BDF2 schemes. Furthermore, we simplify classical numerical analysis of projection methods, allowing us to remove some unnecessary assumptions, such as convexity of the domain. Several numerical experiments are given which verify the convergence rate, and show that projection methods with large grad–div stabilization parameters can dramatically improve accuracy. [ABSTRACT FROM AUTHOR]

We consider the stability of an efficient Crank-Nicolson-Adams-Bashforth method in time, finite element in space, discretization of the Leray- α model. We prove finite-time stability of the scheme in L2, H1, and H2, as well as the long-time L-stability of the scheme under a Courant-Freidrichs-Lewy (CFL)-type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155-1183, 2016 [ABSTRACT FROM AUTHOR]

We investigate a flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for velocity, u | ∂ Ω = g , for use with finite element methods for incompressible flow problems that strongly enforce mass conservation. Typical enforcement via nodal interpolation is not flux-preserving in general, and it can create divergence error even when divergence-free elements are used. We show with analysis and numerical tests that by slightly (and locally) changing nodal interpolation, the enforcement recovers flux-preservation, is optimally accurate, and delivers divergence-free solutions when used with divergence-free finite elements. [ABSTRACT FROM AUTHOR]

This paper introduces a new, reduced-order NS-α (rNS-α) model for the purpose of efficient, stable, and accurate simulations of incompressible flow problems on coarse meshes. We motivate the new model by discussing the difficulties in efficient and stable algorithm construction for the usual NS-α model, and then derive rNS-α by using deconvolution as an approximation to the filter inverse, which reduces the fourth order NS-α formulation to a second order model. After proving the new model is well-posed, we propose a C0 finite element spatial discretization together with an IMEX BDF2 timestepping to create a linearized algorithm that decouples the conservation of mass and momentum equations from the filtering. We rigorously prove the algorithm is well-posed, and provided a very mild timestep restriction, is also stable and converges optimally to the model solution. Finally, we give results of several benchmark computations that confirm the theory and show the proposed model/scheme is effective at efficiently finding accurate coarse mesh solutions to flow problems. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad–div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad–div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator’s matrices are identical to those of grad–div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad–div stabilization, and in all cases, solutions are found to be very similar. [Copyright &y& Elsevier]

Abstract: We study a trapezoidal-in-time, finite-element-in-space discretization of a new Leray regularization model that locally chooses the filtering radius using a deconvolution based indicator function to identify regions where regularization is needed. Because this indicator function is mathematically based, it allows us to establish a rigorous analysis of the resulting numerical algorithm. We prove well-posedness, unconditional stability, and convergence of the proposed algorithm, and test the model on several benchmark problems. [Copyright &y& Elsevier]

Abstract: A deconvolution enhancement of the Navier–Stokes-αβ model for turbulent flow is introduced. The energy and energy-dissipation rate for the enhanced model are derived. It is also shown that the consistency error, relative to the Navier–Stokes equations, and the microscale of the enhanced model are less than those of the Navier–Stokes-αβ model. The proposed model is used to simulate the Taylor–Green vortex problem and results show a qualitatively improved representation of the mean-square vorticity when compared to the Navier–Stokes-αβ model. Numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three-dimensional turbulent flows with Re =200 are used to explore the utility of the model. A benchmark problem of a two-dimensional channel flow over a step for Re =600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions. [Copyright &y& Elsevier]

Abstract: We study a finite element scheme for the 3D Navier–Stokes equations (NSE) that globally conserves energy and helicity and, through the use of Scott–Vogelius elements, enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also show the method can be efficiently computed by exploiting a connection between this method, its associated penalty method, and the method arising from using grad-div stabilized Taylor–Hood elements. Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme. [Copyright &y& Elsevier]

Abstract: It was recently proven in Case et al. (2010) that, under mild restrictions, grad-div stabilized Taylor–Hood solutions of Navier–Stokes problems converge to the Scott–Vogelius solution of that same problem. However, even though the analytical rate was only shown to be (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to . We prove herein the analytical rate is indeed , and extend the result to other incompressible flow problems including Leray-α and MHD. Numerical results are given that verify the theory. [Copyright &y& Elsevier]

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-αsuffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the velocity error that arises from the (necessary) use of the rotational form nonlinearity. The proposed scheme “fixes” these two numerical issues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We prove the scheme is stable, optimally convergent, and the effect of the pressure error on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Abstract: For the three-dimensional incompressible Navier–Stokes equations, we present a formulation featuring velocity, vorticity and helical density as independent variables. We find the helical density can be observed as a Lagrange multiplier corresponding to the divergence-free constraint on the vorticity variable, similar to the pressure in the case of the incompressibility condition for velocity. As one possible practical application of this new formulation, we consider a time-splitting numerical scheme based on an alternating procedure between vorticity–helical density and velocity–Bernoulli pressure systems of equations. Results of numerical experiments include a comparison with some well-known schemes based on pressure–velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation. [Copyright &y& Elsevier]

We analyze the mathematical and physical properties of, and present numerical experiments for, the recently proposed NS-α-deconvolution model of fluid turbulence. This family of models has the well-known NS-α model as its zeroth-order model, and applies the Nth van Cittert deconvolution operator to the filtered terms in NS-α to create the Nth-order NS-α-deconvolution model. This model is proposed in Ref. 29, where it is shown to have consistency error O(α2N + 2). Herein we prove that the model admits unique regular solutions, is frame invariant, and improves the consistency error of NS-α from O(α2) to O(α2N + 2) while requiring (N + 1)1/2 times more (typically 1 ≤ N ≤ 5) degrees of freedom for complete resolution. Numerical experiments show that adding approximate deconvolution to NS-α significantly improves accuracy in computations. [ABSTRACT FROM AUTHOR]

We present enhanced physics-based finite element schemes for two families of turbulence models, the NS-ω models and the Stolz-Adams approximate deconvolution models. These schemes are delicate extensions of a method created for the Navier-Stokes equations in Rebholz (2007) that achieve high physical fidelity by admitting balances of both energy and helicity that match the true physics. The schemes’ development requires carefully chosen discrete curl, discrete Laplacian, and discrete filtering operators, in order to permit the necessary differential operator commutations. [ABSTRACT FROM AUTHOR]

Abstract: This paper gives a comprehensive numerical analysis of a multiscale method for equilibrium Navier Stokes equations. The method includes pressure regularization and eddy viscosity stabilizations both acting only on the finest scales. This method allows for equal order velocity–pressure spaces as well as the linear constant pair and the usual (P k , P k−1) pair. We show the method is optimal in a natural energy norm for all of these pairs of spaces, and provide guidance in choosing the regularization parameters. [Copyright &y& Elsevier]

Algebraic splitting methods are a common approach to solving the saddle point linear systems that arise at each time step of an incompressible flow simulation. There are two main classes of these methods, those of Yosida-type and those of Algebraic Chorin Temam (ACT)-type, with the Yosida methods being predominantly used in practice. We show herein, through new analysis and extensive numerical testing, that ACT methods that include the viscous term stiffness matrix in the modified A-block are unconditionally stable and can be superior to Yosida methods in a range of problems. Particular situations where the ACT-type solvers are advantageous include problems where numerical stability is a concern, as well as problems where strong enforcement of the divergence constraint is important. • ACT splitting can preserve the unconditional stability of BDF2-FEM and BE-FEM methods. • Analysis includes incremental formulation and pressure correction for higher accuracy. • We show ACT methods preserve the FEM divergence constraint. • Our analysis is in the function space setting with appropriate norms. • We give results of several numerical tests that compare Yosida and ACT. [ABSTRACT FROM AUTHOR]

Abstract: This report investigates a Smagorinsky type eddy viscosity model recently proposed by Vreman (2003) in [4]. The model employs a filter operation to represent the small resolved scales and, similar to variational multiscale methods, uses only these small resolved scales to describe the effect of the unresolved scales on the resolved ones. We prove well-posedness of the model, and formulate, analyze and test finite element algorithms for it. [Copyright &y& Elsevier]

Abstract: We investigate numerically a recently proposed vorticity based formulation of the incompressible Navier–Stokes equations. The formulation couples a velocity–pressure system with a vorticity–helicity system, and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties. For a few benchmark problems, we study the performance of a finite element method for this formulation and compare it with the commonly used velocity–pressure based finite element method. It is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems. Further, we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers. [Copyright &y& Elsevier]

Abstract: We study Voigt regularizations for the Navier–Stokes equations (NSEs) and magnetohydrodynamic (MHD) equations in the presence of physical boundary conditions. In particular, we develop the first finite element numerical algorithms for these systems, prove stability and convergence of the algorithms, and test them computationally on problems of practical interest. It is found that unconditionally stable implementations of the Voigt regularization can be made from a simple change to existing NSE and MHD codes, and moreover, optimal convergence of the developed algorithms’ solutions to physical solutions can be obtained if lower-order mixed finite elements are used. Finally, we show that for several benchmark problems, the Voigt regularization on a coarse mesh produces good approximations to NSE and MHD systems; that is, the Voigt regularization provides accurate reduced order models for NSE and MHD flows. [Copyright &y& Elsevier]

Abstract: We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott–Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers. [Copyright &y& Elsevier]

Abstract: We investigate numerical algorithms for viscoelastic fluid flows with defective boundary conditions, where only flow rates or mean pressures are prescribed on parts of the boundary. The defective boundary condition problem is formulated as a minimization problem, where we seek boundary conditions of the flow equations which yield an optimal functional value. Two different approaches are considered in developing computational algorithms for the constrained optimization problem, and results of numerical experiments are presented to compare performance of the algorithms. [Copyright &y& Elsevier]

Abstract: We present a numerical study of the NS-αβ model, which is a recently proposed multiscale variation of the NS-α model that attempts to recapture scales lost through over-regularization by separately modeling dissipation-range scales. We develop a similarity theory for the new model which shows that it is better equipped than the NS-α model to capture smaller-scale behavior. Next, we propose and study an unconditionally stable, optimally accurate, and efficient finite-element implementation for the NS-αβ model; rigorous proofs for stability and convergence are provided. Finally, we present results from two numerical experiments that demonstrate the advantages of the NS-αβ model over the NS-α model. [Copyright &y& Elsevier]

Abstract: Experience with fluid-flow simulation suggests that, in some instances, under-resolved direct numerical simulation (DNS), without a residual-stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales, may be as reliable as simulations based on more complex models of turbulence. One efficient and versatile manner to selectively damp under-resolved spatial scales is by a relaxation regularization, e.g. Stolz and Adams [S. Stolz, N.A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids II (1999) 1699–1701]. We consider the analogous approach based on time scales, time filtering and damping of under-resolved temporal features. The paper explores theoretical and practical aspects of temporally damped fluid-flow simulations. We prove existence of solutions to the resulting continuum model. We also establish the effect of the damping of under-resolved temporal features as the energy balance and dissipation and prove that the time fluctuations→0 in a precise sense. The method is then demonstrated to obtain both steady-state and time-dependent coarse-grid solutions of the Navier–Stokes equations. [Copyright &y& Elsevier]

We consider a fully discrete, effcient algorithm for magnetohydrodynamic (MHD) ow that is based on the Elsässer variable formulation and a timestepping scheme that decouples the MHD system but still provides unconditional stability with respect to the timestep. We prove stability and optimal convergence of the scheme, and also connect the scheme to one based on handling each decoupled system with a penalty-projection method. Numerical experiments are given which verify all predicted convergence rates of our analysis on some analytical test problems, show the results of the scheme on a set of channel ow problems match well the results found when the computation is done with MHD in primitive variable, and finally show the scheme performs well on a channel ow over a step. [ABSTRACT FROM AUTHOR]

Abstract: We study a fluid–flow regularization based on the Leray- model that uses deconvolution in the nonlinear term and dissipation scale modeling in the viscous term. In particular, we establish that this ‘Leray--deconvolution model’ has an energy cascade with an enhanced energy dissipation that enlarges the microscale of the model relative to the Kolmogorov microscale, but captures more of the small scales than does the Leray- model. These theoretical results are confirmed via numerically determined energy spectra. We also propose and analyze an efficient finite-element algorithm method for the proposed model. In addition to establishing stability of the method, the essential ingredient for any numerical study, we demonstrate convergence to a Navier–Stokes solution. A numerical experiment for the two-dimensional flow around an obstacle is also discussed. Results show that enhancing the Leray- model with deconvolution and dissipation scale modeling can significantly increase accuracy. [Copyright &y& Elsevier]

Abstract: We show the velocity solutions to the convective, skew-symmetric, and rotational Galerkin finite element formulations of the Navier–Stokes equations are identical if Scott–Vogelius elements are used, and thus all three formulations will be the same pointwise divergence free solution velocity. A connection is then established between the formulations for grad-div stabilized Taylor–Hood elements: under mild restrictions, the formulations’ velocity solutions converge to each other (and to the Scott–Vogelius solution) as the stabilization parameter tends to infinity. Thus the benefits of using Scott–Vogelius elements can be obtained with the less expensive Taylor–Hood elements, and moreover the benefits of all the formulations can be retained if the rotational formulation is used. Numerical examples are provided that confirm the theory. [ABSTRACT FROM AUTHOR]

Abstract: We study stability, accuracy and efficiency of algorithms for a new regularization of the NSE, the NS- model (in which the vorticity term, , is averaged) given by This is similar to the NS- model (in which the nonlinear term is ), but the small difference opens attractive algorithmic possibilities. We give tests both confirming the predicted rates of convergence and exhibiting some shared limitations of both models. The experiments also show the discrete NS- simulation has greater accuracy () at less cost () and requires significantly fewer degrees of freedom (Section ) than a comparable NS- simulation. The experiments suggest consideration of adding grad–div stabilization and higher accuracy NS--deconvolution models as a next logical step. In fact, this combination produced accurate results (see ) on the coarsest mesh (mesh 1) upon which all other methods, models and variants tested failed. [Copyright &y& Elsevier]