The least-squares pseudo-spectral method for Navier–Stokes equations

First-order system least squares (FOSLS) is a methodology that offers an alternative to standard methods for solving partial differential equations. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. In a companion paper [M. Berndt, T. A. Manteuffel, S. F. McCormick, and G. Starke, Analysis of first-order system least squares ({FOSLS}) for elliptic problems with discontinuous coefficients: Part I, SIAM J. Numer. Anal., 43 (2005), pp. 386--408], ellipticity of an appropriately scaled least-squares bilinear form is established in a natural norm. For some geometries this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross-points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. This paper describes a method for including discrete versions of the singular basis functions together with standard finite element spaces in a least-squares format at little additional computational cost. The singular basis functions are constructed to match the jump conditions that arise at interfaces between regions of continuity of the diffusion coefficient. Because these basis functions must be approximated in practice, the resulting discretization is by nature nonconforming. This necessitates the establishment here of a general error estimate for FOSLS L2 minimization problems discretized by nonconforming finite elements. An advantage of the FOSLS formulation is that this estimate does not involve the consistency error term usually present in bounds for other methods. Based on this general estimate, error bounds are derived for the finite element space that includes singular basis functions. Numerical tests are included that confirm these discretization error bounds. Finally, a multilevel method is developed for solving the discrete system that uses singular basis functions on all levels, and its efficiency is demonstrated by the numerical results.

A stabilized finite volume method for solving the transient Navier---Stokes equations is developed and studied in this paper. This method maintains conservation property associated with the Navier---Stokes equations. An error analysis based on the variational formulation of the corresponding finite volume method is first introduced to obtain optimal error estimates for velocity and pressure. This error analysis shows that the present stabilized finite volume method provides an approximate solution with the same convergence rate as that provided by the stabilized linear finite element method for the Navier---Stokes equations under the same regularity assumption on the exact solution and a slightly additional regularity on the source term. The stability and convergence results of the proposed method are also demonstrated by the numerical experiments presented.

The paper is concerned with the unconditional stability and convergence of characteristics type methods for the time-dependent Navier---Stokes equations. We present optimal error estimates in $$L^2$$L2 and $$H^1$$H1 norms for a typical modified characteristics finite element method unconditionally, while all previous works require certain time-step restrictions. The analysis is based on an iterated characteristic time-discrete system, with which the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the difference between the numerical solution and the solution of the time-discrete system is $$\tau $$?-independent, where $$\tau $$? denotes the time stepsize. Thus numerical solution in $$W^{1,\infty }$$W1,? is bounded and optimal error estimates can be obtained in a traditional way. Numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time-dependent Navier---Stokes equations. The approach used in this paper can be easily extended to many other characteristics-based methods.

SUMMARY We present an inversion algorithm tailored for point gravity data. As the data are from multiple surveys, it is inconsistent with regards to spacing and accuracy. An algorithm design objective is the exact placement of gravity observations to ensure no interpolation of the data is needed prior to any inversion. This is accommodated by discretization using an unstructured tetrahedral finite-element mesh for both gravity and density with mesh nodes located at all observation points and a first-order system least-squares (FOSLS) formulation for the gravity modelling equations. Regularization follows the Bayesian framework where we use a differential operator approximation of an exponential covariance kernel, avoiding the usual requirement of inverting large dense covariance matrices. Rather than using higher order basis functions with continuous derivatives across element faces, regularization is also implemented with a FOSLS formulation using vector-valued property function (density and its gradient). Minimization of the cost function, comprised of data misfit and regularization, is achieved via a Lagrange multiplier method with the minimum of the gravity FOSLS functional as a constraint. The Lagrange variations are combined into a single equation for the property function and solved using an integral form of the pre-conditioned conjugate gradient method (I-PCG). The diagonal entries of the regularization operator are used as the pre-conditioner to minimize computational costs and memory requirements. Discretization of the differential operators with the finite-element method (FEM) results in matrix systems that are solved with smoothed aggregation algebraic multigrid pre-conditioned conjugate gradient (AMG-PCG). After their initial setup, the AMG-PCG operators and coarse grid solvers are reused in each iteration step, further reducing computation time. The algorithm is tested on data from 23 surveys with a total of 6519 observation points in the Mt Isa–Cloncurry region in north–west Queensland, Australia. The mesh had about 2.5 million vertices and 16.5 million cells. A synthetic case was also tested using the same mesh and error measures for localized concentrations of high and low densities. The inversion results for different parameters are compared to each other as well as to lower order smoothing. Final inversion results are shown with and without depth weighting and compared to previous geological studies for the Mt Isa–Cloncurry region.

Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier---Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells (SIAM J Sci Comput 34(2):A889---A913, 2012). We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.

Stabilized multiscale finite element method for the stationary Navier–Stokes equations

Optimal orders of convergence for the Fourier-Galerkin method for two-dimensional Navier–Stokes equations in various energy norms and in $L^p$-norm of the velocity field are proved. Optimal orders of convergence for the Fourier-collocation method are also proved. Extensions of these results to three-dimensional Navier–Stokes equations in the original form and the rotational form are presented. Semigroup formulations and the variation of constants formula are the essential tools in the analysis.

We propose a set of generalized incompressible fluid dynamical equations, which interpolates between the Burgers and Navier–Stokes equations in two dimensions and study their properties theoretically and numerically. It is well-known that under the assumption of potential flows the multi-dimensional Burgers equations are integrable in the sense they can be reduced to the heat equation, via the so-called Cole-Hopf linearization. On the other hand, it is believed that the Navier–Stokes equations do not possess such a nice property. Take, for example, the 2D Navier–Stokes equations and rotate the velocity gradient by 90 degrees, we then obtain a system which is equivalent to the Burgers equations. Based on this observation, we introduce a system of generalized incompressible fluid dynamical equations by rotating velocity gradient through a continuous angle parameter. That way we can compare properties of an integrable system with those of non-integrable ones by relating them through a continuous parameter. Using direct numerical experiments we show how the flow properties change (actually, deteriorate) when we increase the angle parameter α from 0 to π / 2 . It should be noted that the case associated with least regularity, namely the Burgers equations ( α = π / 2 ), is integrable via the heat kernel. We also formalize a perturbative treatment of the problem, which in principle yields the solution of the Navier–Stokes equations on the basis of that of the Burgers equations. The principal variation around the Burgers equations is computed numerically and is shown to agree well with the results of direct numerical simulations for some time.

The Incompressible Navier-Stokes Equations (NSEs) are on the list of Millennium Problems, to prove their existence and uniqueness of solutions. The NSEs can be formulated in a periodic 3D domain, where they are termed the Periodic Navier Stokes (PNS) Equations, and can be treated on a subspace spanning a 3-dimensional torus, or T3. Treating the PNS Equations in T3-space, this article demonstrates that a decaying of turbulence occurs in the 3D case for the z component of velocity when non-smooth initial conditions are considered for x, y components of velocity and that ‘vorticity’ sheets in the small scales of 3D turbulence dominate the flow to the extent that non-smooth temporal solutions exist for the z velocity for smooth initial data for the x, y components of velocity. Unlike the Navier-Stokes equations, which have no anti-symmetric vorticity tensor, there are new governing equations which have vorticity tensor and can be decomposed into a rotational part(Liutex), antisymmetric shear and compression and stretching. It is shown that under these recent findings, that there is a strong correlation between vorticity and vorticies for (PNS).

High-order arbitrary Lagrangian–Eulerian discontinuous Galerkin methods for the incompressible Navier–Stokes equations

We prove that a space-time hybridized discontinuous Galerkin method for the evolutionary Navier–Stokes equations converges to a weak solution as the time step and mesh size tend to zero. Moreover, we show that this weak solution satisfies the energy inequality. To perform our analysis, we make use of discrete functional analysis tools and a discrete version of the Aubin–Lions–Simon theorem.

A defect-correction stabilized finite element method for Navier–Stokes equations with friction boundary conditions

: This final report on AFOSR project 'Multilevel Techniques in Large Scale Computation' (AFOSR F49620-92-J-0439) is described according to the following topics, which include examples of how these applications have led to important basic advances: first-order system least squares (FOSLS); electromagnetics and seismology; meteorology; multiscale atmospheric data assimilation; molecular dynamics; robust three-dimensional solvers; and parallel adaptive refinement.

In this paper, we analyze Nitsche’s method for the stationary Navier–Stokes equations on Lipschitz domains under minimal regularity assumptions. Our analysis provides a robust formulation for implementing slip (i.e., Navier) boundary conditions in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using the Banach Nečas–Babuška and Banach fixed point theorems under standard small data assumptions. We also provide optimal convergence rates for the approximation error. Furthermore, we propose a quasi-static VMS-LES formulation with Nitsche for the Navier–Stokes equations with slip boundary conditions to address the simulation of incompressible fluids at high Reynolds numbers. We validate our theory through several numerical tests in well-established benchmark problems.