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The incremental Picard Yosida (IPY) method has recently been developed as an iteration for nonlinear saddle point problems that is as effective as Picard but more efficient. By combining ideas from algebraic splitting of linear saddle point solvers with incremental Picard‐type iterations and grad‐div stabilization, IPY improves on the standard Picard method by allowing for easier linear solves at each iteration—but without creating more total nonlinear iterations compared to Picard. This paper extends the IPY methodology by studying it together with Anderson acceleration (AA). We prove that IPY for Navier–Stokes and regularized Bingham fits the recently developed analysis framework for AA, which implies that AA improves the linear convergence rate of IPY by scaling the rate with the gain of the AA optimization problem. Numerical tests illustrate a significant improvement in convergence behavior of IPY methods from AA, for both Navier–Stokes and regularized Bingham. [ABSTRACT FROM AUTHOR]

A one-step analysis of Anderson acceleration with general algorithmic depths is presented. The resulting residual bounds within both contractive and noncontractive settings reveal the balance between the contributions from the higher and lower order terms, which are both dependent on the success of the optimization problem solved at each step of the algorithm. The new residual bounds show the additional terms introduced by the extrapolation produce terms that are of a higher order than was previously understood. In the contractive setting these bounds sharpen previous convergence and acceleration results. The bounds rely on sufficient linear independence of the differences between consecutive residuals, rather than assumptions on the boundedness of the optimization coefficients, allowing the introduction of a theoretically sound safeguarding strategy. Several numerical tests illustrate the analysis primarily in the noncontractive setting, and demonstrate the use of the method, the safeguarding strategy and theory-based guidance on dynamic selection of the algorithmic depth, on a p-Laplace equation, a nonlinear Helmholtz equation and the steady Navier–Stokes equations with high Reynolds number in three spatial dimensions. [ABSTRACT FROM AUTHOR]

We introduce, analyze, and test an interpolation operator designed for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of the L2 projection operator onto piecewise constant functions on a coarse mesh, but which allows nudging to be done completely at the linear algebraic level, independent of the rest of the discretization, with a diagonal matrix that is simple to construct; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient stability and accuracy properties, and we apply it to algorithms for both fluid transport DA and incompressible Navier–Stokes DA. For both applications we prove the DA solutions with arbitrary initial conditions converge to the true solution (up to discretization error) exponentially fast in time, and are thus long‐time accurate. Results of several numerical tests are given, which both illustrate the theory and demonstrate its usefulness on practical problems. [ABSTRACT FROM AUTHOR]

We consider local balances of momentum and angular momentum for the incompressible Navier–Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier–Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory. [ABSTRACT FROM AUTHOR]

We propose new, efficient, and simple nonlinear iteration methods for the incompressible Navier-Stokes equations. The methods are constructed by applying Yosida-type algebraic splitting to the linear systems that arise from grad-div stabilized finite element implementations of incremental Picard and Newton iterations. They are efficient because at each nonlinear iteration, the same symmetric positive definite Schur complement system needs to be solved, which allows for CG to be used for inner and outer solvers, simple preconditioning, and the reusing of preconditioners. For the proposed incremental Picard-Yosida and Newton-Yosida iterations, we prove under small data conditions that the methods converge to the solution of the discrete nonlinear problem. Numerical tests are performed which illustrate the effectiveness of the method on a variety of test problems. [ABSTRACT FROM AUTHOR]

We propose, analyze, and test Anderson-accelerated Picard iterations for solving the incompressible Navier-Stokes equations (NSE). Anderson acceleration has recently gained interest as a strategy to accelerate linear and nonlinear iterations, based on including an optimization step in each iteration. We extend the Anderson acceleration theory to the steady NSE setting and prove that the acceleration improves the convergence rate of the Picard iteration based on the success of the underlying optimization problem. The convergence is demonstrated in several numerical tests, with particularly marked improvement in the higher Reynolds number regime. Our tests show it can be an enabling technology in the sense that it can provide convergence when both usual Picard and Newton iterations fail. [ABSTRACT FROM AUTHOR]

Abstract We study numerical schemes for incompressible Navier–Stokes equations using IMEX temporal discretizations, finite element spatial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani et al. (2014)). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical. [ABSTRACT FROM AUTHOR]

We study well-posedness of a velocity-vorticity formulation of the Navier–Stokes equations, supplemented with no-slip velocity boundary conditions, a corresponding zero-normal condition for vorticity on the boundary, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier–Stokes equations. [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]

In this paper, we investigate theoretically and computationally the conservation properties of reduced order models (ROMs) for fluid flows. Specifically, we investigate whether the ROMs satisfy the same (or similar) energy balance and mass conservation as those satisfied by the Navier–Stokes equations. All of our theoretical findings are illustrated and tested in numerical simulations of a 2D flow past a circular cylinder at a Reynolds number R e = 100 . First, we investigate the ROM energy balance. We show that using the snapshot average for the centering trajectory (which is a popular treatment of nonhomogeneous boundary conditions in ROMs) yields an incorrect energy balance. Then, we propose a new approach, in which we replace the snapshot average with the Stokes extension. Theoretically, the Stokes extension produces an accurate energy balance. Numerically, the Stokes extension yields more accurate results than the standard snapshot average, especially for longer time intervals. Our second contribution centers around ROM mass conservation. We consider ROMs created using two types of finite elements: the standard Taylor–Hood (TH) element, which satisfies the mass conservation weakly, and the Scott–Vogelius (SV) element, which satisfies the mass conservation pointwise. Theoretically, the error estimates for the SV-ROM are sharper than those for the TH-ROM. Numerically, the SV-ROM yields significantly more accurate results, especially for coarser meshes and longer time intervals. [ABSTRACT FROM AUTHOR]

We consider two modifications of the Arrow–Hurwicz (AH) iteration for solving the incompressible steady Navier–Stokes equations for the purpose of accelerating the algorithm: grad–div stabilization, and Anderson acceleration. AH is a classical iteration for general saddle point linear systems and it was later extended to Navier–Stokes iterations in the 1970's which has recently come under study again. We apply recently developed ideas for grad–div stabilization and divergence-free finite element methods along with Anderson acceleration of fixed point iterations to AH in order to improve its convergence. Analytical and numerical results show that each of these methods improves AH convergence, but the combination of them yields an efficient and effective method that is competitive with more commonly used solvers. [ABSTRACT FROM AUTHOR]

This paper studies a new alteration of Yosida algebraic splitting methods for the Navier{Stokes equations. By applying the usual or pressure-corrected Yosida splitting techniques to discretizations written in terms of velocity and pressure updates (un+1 -- un; pn+1, pn), we show that the accuracy is increased by one full order in Δt without any additional cost in the respective methods. Proofs of the convergence results are given both in linear algebraic and finite element frameworks. Several numerical tests are given which reveal the (sometimes dramatic) improvement in accuracy offered by the proposed fix. [ABSTRACT FROM AUTHOR]

The divergence constraint of the incompressible Navier{Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which in uences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using nonrobust discretizations and the improvements obtained by utilizing pressure-robust discretizations. [ABSTRACT FROM AUTHOR]

We present an analytical and numerical study of the reduced NS- α (rNS- α ) model of incompressible fluid flow, which was recently proposed as an approximation to NS- α that allows for more efficient computations in the C 0 finite element setting. Our analytical study consists of establishing connections between rNS- α to some well-studied models, and determining the model’s integral invariants, energy cascade, and energy dissipation rate. Numerical simulations of turbulent flows with coarse meshes are performed, and excellent results found, for forced homogeneous, isotropic turbulence, and for turbulent channel flow with friction Reynolds numbers Re τ = 395, 590, and 1000. [ABSTRACT FROM AUTHOR]

We present a numerical study of the sensitivity of the grad-div stabilization parameter for mixed finite element discretizations of incompressible flow problems. For incompressible isothermal and non-isothermal Stokes equations and Navier-Stokes equations, we develop the associated sensitivity equations for changes in the grad-div parameter. Finite element schemes are devised for computing solutions to the sensitivity systems, analyzed for stability and accuracy, and finally tested on several benchmark problems. Our results reveal that solutions are most sensitive for small values of the parameter, near obstacles and corners, when the pressure is large, and when the viscosity is small. [ABSTRACT FROM AUTHOR]

We investigate both theoretically and numerically the consistency between the nonlinear discretization in full order models (FOMs) and reduced order models (ROMs) for incompressible flows. To this end, we consider two cases: (i) FOM–ROM consistency, i.e., when we use the same nonlinearity discretization in the FOM and ROM; and (ii) FOM–ROM inconsistency, i.e., when we use different nonlinearity discretizations in the FOM and ROM. Analytically, we prove that while the FOM–ROM consistency yields optimal error bounds, FOM–ROM inconsistency yields additional terms dependent on the FOM divergence error, which prevent the ROM from recovering the FOM as the number of modes increases. Computationally, we consider channel flow around a cylinder and Kelvin–Helmholtz instability, and show that FOM–ROM consistency yields significantly more accurate results than FOM–ROM inconsistency. [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]

We study mathematical and physical properties of a family of recently introduced, reduced-order approximate deconvolution models. We first show a connection between these models and the Navier-Stokes-Voigt model, and also that Navier-Stokes-Voigt can be re-derived in the approximate deconvolution framework. We then study the energy balance and spectra of the model, and provide results of some turbulent-flow computations that backs up the theory. Analysis of global attractors for the model is also provided, as is a detailed analysis of the Voigt model's treatment of pulsatile flow. [ABSTRACT FROM AUTHOR]

This paper presents newanalytical results and the first numerical results for a recently proposedmultiscale deconvolution model (MDM) recently proposed. The model involves a large-eddy simulation closure that uses a novel deconvolution approach based on the introduction of two distinct filtering length scales.We establish connections between theMDMand two other models, and, on the basis of one of these connections, we establish an improved regularity estimate for MDM solutions.We also prove that theMDMpreserves Taylor-eddy solutions of theNavier-Stokes equations and therefore does not distort this particular vortex structure. Simulations of the MDM are performed to examine the accuracy of the MDM and the effect of the filtering length scales on energy spectra for three-dimensional homogeneous and isotropic flows. Numerical evidence for all tests clearly indicates that the MDM gives very accurate coarse-mesh solutions and that this multiscale approach to deconvolution is effective. [ABSTRACT FROM AUTHOR]

This paper analyzes the accuracy of the ‘discretize-then-split’ Yosida solver for incompressible flow problems, when divergence-free elements are used together with grad-div stabilization (with parameter γ ). The Yosida method uses an inexact block LU factorization to create linear algebraic systems that are easier to solve, but at the expense of accuracy. We prove the difference between solutions of the exact and approximated linear algebraic systems is O ( γ − 2 ) in the natural norms of the associated finite element problems, and thus that full accuracy can be obtained by the Yosida method if large γ is used ( γ ≥ 10 is sufficient in our numerical examples). The proof is based on transforming the Yosida inexact linear algebraic system into finite element problems, and analyzing these problems with finite element techniques based on pointwise divergence-free subspaces and their orthogonal complements. [ABSTRACT FROM AUTHOR]

We prove a long-time stability result for the finite element in space, linear extrapolated Crank–Nicolson in time discretization of the Navier–Stokes equations (NSE). From this result and a numerical experiment, we show the importance of discrete mass conservation in long-time simulations of the NSE. That is, we show that using elements that strongly enforce mass conservation can provide significantly more accurate solutions over long times, compared to those that enforce it weakly. [ABSTRACT FROM AUTHOR]

We prove conservation of a regularized helicity H L : = ∫ Ω u ⋅ w d x for the Leray model (and its variants) of turbulent flow, where w is the solution of a Leray-regularized vorticity equation. The usual definition of helicity is H = ∫ Ω u ⋅ ( ∇ × u ) d x , which is considered by Navier–Stokes flows, but is not a conserved quantity of the Leray model. However, if u is a Leray solution, then the difference between H and H L is that H L uses a regularized vorticity and H uses the curl of a regularized velocity. The results are extended to show that the standard Crank–Nicolson finite element method for Leray models conserves both discrete energy and discrete regularized helicity. [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]

We study analytically and numerically the relaxation time of flow evolution governed by the Navier-Stokes-Voigt (NSV) model. We first show that for the Taylor–Green vortex decay problem, NSV admits an exact solution which evolves slower than true fluid flow. Secondly, we show numerically for a channel flow test problem using standard discretisation methods that although NSV provides more regular solutions compared to usual Navier-Stokes solutions, NSV approximations take significantly longer to reach the steady state. [ABSTRACT FROM AUTHOR]

We consider a pressure correction temporal discretization for the incompressible Navier-Stokes equations in EMAC form. We prove stability and error estimates for the case of mixed finite element spatial discretization, and in particular that the Gronwall constant's exponential dependence on the Reynolds number is removed (for sufficiently smooth true solutions) or at least significantly reduced compared to the commonly used skew-symmetric formulation. We also show the method preserves momentum and angular momentum, and while it does not preserve energy it does admit an energy inequality. Several numerical tests show the advantages EMAC can have over other commonly used formulations of the nonlinearity. Additionally, we discuss extensions of the results to the usual Crank-Nicolson temporal discretization. [ABSTRACT FROM AUTHOR]

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]

This article studies two methods for obtaining excellent mass conservation in finite element compuations of the Navier Stokes equations using continuous velocity fields. With a partitular mesh construction, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that the limit of the grad-div stabilized Taylor Hood solutions to the Navier-Stokes problem converges to the Scott Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations. [ABSTRACT FROM AUTHOR]

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]

We study an efficient finite element method for the NS-ω model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence. Additionally, an improved choice of filtering radius (versus the usual choice of the average mesh width) for the scheme is found, by identifying a connection with a scheme for the velocity-vorticity-helicity NSE formulation. Several numerical experiments are given that demonstrate the performance of the scheme, and the improvement offered by the new choice of the filtering radius. [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]

Abstract: We study the helicity balance of the Galerkin method for the 3D Navier–Stokes equations, and show that although it does not appear to correctly balance helicity in the usual sense, it instead admits a slightly altered helicity balance that matches that of the underlying physics, up to boundary conditions. [Copyright &y& Elsevier]

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]

This paper considers an enhancement of the classical iterated penalty Picard (IPP) method for the incompressible Navier–Stokes equations, where we restrict our attention to O (1) penalty parameter, and Anderson acceleration (AA) is used to significantly improve its convergence properties. After showing the fixed point operator associated with the IPP iteration is Lipschitz continuous and Lipschitz continuously (Frechet) differentiable, we apply a recently developed general theory for AA to conclude that IPP enhanced with AA improves its linear convergence rate by the gain factor associated with the underlying AA optimization problem. Results for several challenging numerical tests are given and show that IPP with penalty parameter 1 and enhanced with AA is a very effective solver. • With Anderson acceleration, iterated penalty for NSE is effective with O(1) penalty. • IPP can now be used for problems too large for direct linear solvers. • AA reduces linear convergence rate scaling by optimization problem gain factor • Simulation results show IPP+AA with penalty 1 is a very effective solver. [ABSTRACT FROM AUTHOR]

We present a new finite element scheme for solving the Navier-Stokes equations that exactly conserves both energy (fΩ u²) and helicity (fΩ u · (∇ × u)) in the absence of viscosity and external force. We prove stability, exact conservation, and convergence for the scheme. Energy and helicity are exactly conserved by using a combination of the usual (convective) form with the rotational form of the nonlinearity and solving for both velocity and a projected vorticity in a trapezoidal time discretization. Numerical results are presented that compare the scheme to the usual trapezoidal schemes. [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]

In this paper, we consider the recently introduced EMAC formulation for the incompressible Navier–Stokes (NS) equations, which is the only known NS formulation that conserves energy, momentum and angular momentum when the divergence constraint is only weakly enforced. Since its introduction, the EMAC formulation has been successfully used for a wide variety of fluid dynamics problems. We prove that discretizations using the EMAC formulation are potentially better than those built on the commonly used skew-symmetric formulation, by deriving a better longer time error estimate for EMAC: while the classical results for schemes using the skew-symmetric formulation have Gronwall constants dependent on exp (C ⋅ R e ⋅ T) with R e the Reynolds number, it turns out that the EMAC error estimate is free from this explicit exponential dependence on the Reynolds number. Additionally, it is demonstrated how EMAC admits smaller lower bounds on its velocity error, since incorrect treatment of linear momentum, angular momentum and energy induces lower bounds for L 2 velocity error, and EMAC treats these quantities more accurately. Results of numerical tests for channel flow past a cylinder and 2D Kelvin–Helmholtz instability are also given, both of which show that the advantages of EMAC over the skew-symmetric formulation increase as the Reynolds number gets larger and for longer simulation times. • We study EMAC formulation for incompressible Navier–Stokes equations. • Superior performance of EMAC often observed but lacks rigorous explanation. • First convergence analysis of EMAC is given, and sheds light on performance. • Error estimate is proved with Gronwall exponent independent of Re number. [ABSTRACT FROM AUTHOR]

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]

We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement data into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs. • Incorporate nudging based data assimilation into a reduced order modeling framework. • Prove that the new DA-ROM converges exponentially fast in time to the true solution. • Demonstrate that the DA-ROM strategy can provide long time accuracy in ROMs. [ABSTRACT FROM AUTHOR]

This paper considers improving the Picard and Newton iterative solvers for the Navier–Stokes equations in the setting where data measurements or solution observations are available. We construct adapted iterations that use continuous data assimilation (CDA) style nudging to incorporate the known solution data into the solvers. For CDA-Picard, we prove the method has an improved convergence rate compared to usual Picard, and the rate improves as more measurement data is incorporated. We also prove that CDA-Picard is contractive for larger Reynolds numbers than usual Picard, and the more measurement data that is incorporated the larger the Reynolds number can be with CDA-Picard still being contractive. For CDA-Newton, we prove that the domain of convergence, with respect to both the initial guess and the Reynolds number, increases as the amount of measurement data is increased. Additionally, for both methods we show that CDA can be implemented as direct enforcement of measurement data into the solution. Numerical results for common benchmark Navier–Stokes tests illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose, analyze and test a post-processing implementation of a projection-based variational multiscale (VMS) method with proper orthogonal decomposition (POD) for the incompressible Navier–Stokes equations. The projection-based VMS stabilization is added as a separate post-processing step to the standard POD approximation, and since the stabilization step is completely decoupled, the method can easily be incorporated into existing codes, and stabilization parameters can be tuned independent from the time evolution step. We present a theoretical analysis of the method, and give results for several numerical tests on benchmark problems which both illustrate the theory and show the proposed method’s effectiveness. [ABSTRACT FROM AUTHOR]

We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping methods for the Navier–Stokes equations (NSE). The method combines a linear treatment of the advection term, together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. We rigorously prove that the entire new family of methods are unconditionally stable and O ( Δ t 2 ) accurate. The idea of ‘curvature stabilization’ is new to CFD and is intended as an improvement over the commonly used ‘speed stabilization’, which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients. Numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows. [ABSTRACT FROM AUTHOR]

We give a detailed analytical study of a Leray model of incompressible flow that uses nonlinear filtering based on indicator functions. The indicator functions allow for local regularization, instead of global regularization which can over-smooth and dampen out important flow structures. The key to the analysis is the identification of the indicator function as a Nemyskii operator. After proving well-posedness, we provide a numerical study which includes proving optimal convergence of finite element method for the model, as well as several numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: This paper shows that use of a recently introduced sparse grad-div stabilization can increase the accuracy of projection methods for solving the Navier–Stokes equations. Sparse grad-div stabilization has recently been introduced as an alternative to standard grad-div stabilization which has a sparser matrix representation. For both sparse and standard grad-div stabilized projection methods, we prove error estimates and provide numerical experiments which reveal that both stabilizations can cause a significant decrease in the error. We then compare iterative solvers for the linear systems of equations arising from the use of both of the stabilizations. A theoretical analysis of a simplified model problem as well as numerical tests show that iterative solvers perform better for systems arising from sparse grad-div compared to standard grad-div stabilized systems. [Copyright &y& Elsevier]

We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier–Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper (Li et al. 2023), and we extend the methodology by improving the analysis to be in the L 2 norm instead of a weighted H 1 norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration. [ABSTRACT FROM AUTHOR]

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]