A nearly optimal preconditioner for the Navier–Stokes equations

We present a preconditioner for the linearized Navier–Stokes equations which is based on the combination of a fast transform approximation of an advection diffusion problem together with the recently introduced ‘BFBT^T’ preconditioner of Elman (SIAM Journal of Scientific Computing, 1999; 20 :1299–1316). The resulting preconditioner when combined with an appropriate Krylov subspace iteration method yields the solution in a number of iterations which appears to be independent of the Reynolds number provided a mesh Péclet number restriction holds, and depends only mildly on the mesh size. The preconditioner is particularly appropriate for problems involving a primary flow direction. Copyright © 2001 John Wiley & Sons, Ltd.     

A new stabilized finite element method for advection‐diffusion‐reaction equations

In this article, a new stabilized finite element method is proposed and analyzed for advection‐diffusion‐reaction equations. The key feature is that both the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number are reasonably incorporated into the newly designed stabilization parameter. The error estimates are established, where, up to the regularity‐norm of the exact solution, the explicit‐dependence of the diffusivity, advection, reaction, and mesh size (or the dependence of the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number) is revealed. Such dependence in the error bounds provides a mathematical justification on the effectiveness of the proposed method for any values of diffusivity, advection, dissipative reaction, and mesh size. Numerical results are presented to illustrate the performance of the method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 616–645, 2016     

Uniform convergence of discretization error for a singular perturbation problem

Cell-centered discretization of the convection-diffusion equation with large Péclet number Pe is analyzed, in the presence of a parabolic boundary layer. It is shown theoretically how, by suitable mesh refinement in the boundary layer, the accuracy can be made to be uniform in Pe, at the cost of a IogPe increase of the number of grid cells, in the case of upwind discretization. Numerical experiments are presented indicating that this can in practice also be achieved with a Pe-independent number of grid cells, both with upwind and central discretization, and with vertex-centered discretization. © 1996 John Wiley & Sons, Inc.     

A posteriori error estimates for subgrid viscosity stabilized approximations of convection–diffusion equations

We derive a posteriori error estimates for subgrid viscosity stabilized finite element approximations of convection–diffusion equations in the high Péclet number regime. Two estimators are analyzed: an asymptotically robust one and a fully robust one with respect to the Péclet number. Numerical results on test cases with boundary layers or internal layers show that the asymptotically robust estimator can be used to construct adaptive meshes.     

Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

A nonconforming (Crouzeix–Raviart) finite element method with subgrid viscosity is analyzed to approximate advection‐diffusion‐reaction equations. The error estimates are quasi‐optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order in the mesh size for the L²‐norm and optimal for the advective derivative on quasi‐uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Inviscid Incompressible Limits Under Mild Stratification: A Rigorous Derivation of the Euler–Boussinesq System

We consider the full Navier–Stokes–Fourier system in the singular regime of small Mach and large Reynolds and Péclet numbers, with ill prepared initial data on an unbounded domain \(\Omega \subset R^3\) with a compact boundary. We perform the singular limit in the framework of weak solutions and identify the Euler–Boussinesq system as the target problem.     

A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Boundary layers for cellular flows at high Péclet numbers

We analyze the behavior of solutions of steady advection‐diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ? 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(?^1/2), ? = Pe^?1, around the flow separatrices. We construct an ?‐dependent approximate “water pipe problem” purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ?‐independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ?‐independent problem on this graph. Finally, we show that the Dirichlet‐to‐Neumann map of the water pipe problem approximates the Dirichlet‐to‐Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers. © 2004 Wiley Periodicals, Inc.     

On the flow of a viscous incompressible fluid between two coaxial rotating porous cylinders

Exact solution of the Navier-Stokes equations for the laminar flow of a viscous incompressible fluid between two coaxial rotating porous cylinders, kept at constant temperatures, has been studied. The rate of injection at one cylinder is taken to be the same as the rate of suction at the other. Expressions for the velocity and temperature distributions and for the torque required to turn the outer cylinder are obtained. The effects of λ (injection parameter), σ (the ratio of the radii of the cylinders) and Pé (Péclet number = λPr) on them are shown graphically.     

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France.     

Effective models for reactive flow under a dominant Péclet number and order one Damköhler number: Numerical simulations

The paper is devoted to the longitudinal dispersion of a soluble substance released in a steady laminar flow through a slit channel with heterogeneous reaction at the outer wall. The reactive transport happens in the presence of a dominant Péclet number and order one Damköhler number. In particular, these Péclet numbers correspond to Taylor’s dispersion regime. An effective model for the enhanced diffusion in this context was derived recently. It contains memory effects and contributions to the effective diffusion and effective advection velocity, due to the flow and chemistry reaction regime. In the present paper, we show through numerical simulations the efficiency of this new model. In particular, using Taylor’s ‘historical’ parameters, we illustrate that our derived contributions are important and that using them is necessary in order to simulate correctly the reactive flows. We emphasize three main points. First, we show how the effective diffusion is enhanced by chemical effects at dispersive times. Second, our model captures an intermediate regime where the diffusion is anomalous and the distribution is asymmetric. Third, we show how the chemical effects also slow down the average speed of the front.     

Fluctuations of rotational and translational degrees of freedom in an interacting active dumbbell system

We study the dynamical properties of a two-dimensional ensemble of self-propelled dumbbells with only repulsive interactions. After summarizing the behavior of the translational and rotational mean-square displacements in the homogeneous phase that we established in a previous study, we analyze their fluctuations. We study the dependence of the probability distribution functions in terms of the Péclet number, describing the relative role of active forces and thermal fluctuations, and of particle density.     

A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

     

Robust hierarchical a posteriori error estimators for stabilized convection–diffusion problems

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Analytical and numerical solutions to some one-dimensional relativistic heat equations

Analytical and numerical solutions to a family of one-dimensional (nonlinear) relativistic heat equations in finite domains are presented. The analytical solutions correspond to steady state conditions in the absence of source terms and have been obtained as functions of the (absolute) temperature at one of the boundaries, a characteristic exponent, a Péclet number based on the speed of light and the heat flux, while the numerical ones correspond to an initial Gaussian temperature distribution, adiabatic boundary conditions and different values of the Péclet number and a characteristic exponent. It is shown that, for steady conditions, the difference between the (nondimensional) temperature of the relativistic heat equation and that corresponding to Fourier law is very large for large values of both the coefficient and the exponent of the nonlinearity that characterize the relativistic contribution to the heat flux, small values of the temperature at one of the boundaries and large heat fluxes. Travelling-wave solutions of the wave-front type are reported for odd values of the nonlinearity exponent in infinite domains and in the absence of source terms. For an initial Gaussian distribution, it is shown that the relativistic contribution to heat transfer results in the formation of two triangular corner regions where the temperature is equal to the initial one, and the formation of two temperature fronts that propagate towards the domain’s boundaries. The amplitude and steepness of these fronts increase whereas their width and speed decrease as the Péclet number is decreased. It is also shown that the effects of the characteristic exponent are small provided that its value is greater than about two, and that, in the absence of source terms, the temperature becomes uniform in space and constant in time for adiabatic boundary conditions. In the presence of source terms and for adiabatic boundary conditions, it is shown that, soon after the temperature fronts hit the boundaries, the temperature becomes uniform in space but may either increase or decrease with time until it reaches a stable fixed point of the source term. For a cubic source term that exhibits bistability, it is shown that the temperature tends to the attractor of lowest temperature.     

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].     

A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation

We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection–reaction–diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet–Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.     

Asymptotic solution of the problem of heat transfer between a plate and an unbounded uniform fluid flow

The three leding terms of the asymptotic expansion of the solution of the problem of convective heat transfer between a thin plate of finite length and arbitrary surface temperature and an unbounded uniform fluid flow are obtained analytically for low Péclet and Prandtl numbers.     

Multiple solutions of mixed convection boundary-layer approximations in a porous medium

This note deals with a theoretical analysis of the existence, non-uniqueness and non-existence of similarity solutions of the two-dimensional mixed convection boundary-layer flow over a vertical surface with a power law temperature. Here, it is assumed that the surface is embedded in a saturated porous media. The results depend on the power law exponent and the ratio of the Rayleigh to Péclet numbers. It is shown, under certain circumstance, that the problem has an infinite number of solutions.     

Discontinuous Petrov–Galerkin method based on the optimal test space norm for steady transport problems in one space dimension

We revisit the finite element analysis of convection-dominated flow problems within the recently developed Discontinuous Petrov–Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the optimal test space norm. This makes the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. We employ discontinuous piecewise Bernstein polynomials as trial functions and construct a subgrid discretization that accounts for the singular perturbation character of the problem to resolve the corresponding optimal test functions. We also show that a smooth B-spline basis has certain computational advantages in the subgrid discretization. The overall effectiveness of the algorithm is demonstrated on two problems for the linear advection–diffusion equation.