Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703; A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001. J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102.     

Modeling of two-phase flows with surface tension by finite pointset method (FPM)

A meshfree method for two-phase immiscible incompressible flows including surface tension is presented. The continuum surface force (CSF) model is used to include the surface tension force. The incompressible Navier–Stokes equation is considered as the mathematical model. Application of implicit projection method results in linear second-order partial differential equations for velocities and pressure. These equations are then solved by the finite pointset method (FPM), which is a meshfree and Lagrangian method. The fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle. The interface is tracked automatically by advecting the color functions for each particle. Two test cases, Laplace's law and the Rayleigh–Taylor instability in 2D have been presented. The results are found to be consistent with the theoretical results.     

A stream function-vorticity formulation coupled with boundary integrals for the two-dimensional exterior Stokes problem

In this work, we derive a stream function-vorticity variational formulation coupled with boundary integrals for the exterior Stokes problem in two dimensions, when the right-hand side has a bounded support. The stream function-vorticity formulation is expressed in a bounded region containing the support of the right-hand side, and the boundary conditions on the artificial boundary are obtained by an integral representation. We prove that this coupled formulation is equivalent to the original Stokes problem.     

Application of finite element method in aeroelasticity

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil, which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element approximation of the Navier–Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations, stabilization the finite element discretization and coupling of both models is discussed. Moreover, the Reynolds averaged Navier–Stokes (RANS) system of equations together with the Spallart–Almaras turbulence model is also discussed. The computational results of aeroelastic calculations are presented and compared with the NASTRAN code solutions.     

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.     

Bubble finite elements for the primitive equations of the ocean

We introduce in this paper two original Mixed methods for the numerical resolution of the (stationary) Primitive Equations (PE) of the Ocean. The PE govern the behavior of oceanic flows in shallow domains for large time scales. We use a reduced formulation (Lions et al. [28]) involving horizontal velocities and surface pressures. By using bubble functions constructed ad-hoc, we are able to define two stable Mixed Methods requiring a low number of degrees of freedom. The first one is based on the addition of bubbles of reduced support to velocities elementwise. The second one makes use of conic bubbles of extended support along the vertical coordinate. The latter constitutes a genuine mini-element for the PE, e.g., it requires the least number of extra degrees of freedom to stabilize piecewise linear hydrostatic pressures. Both methods verify a specific inf-sup condition and provide stability and convergence. Finally, we compare several numerical features of the proposed pairs in the context of other FE methods found in the literature.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Weighted quadrature rules for finite element methods

We discuss the numerical integration of polynomials times non-polynomial weighting functions in two dimensions arising from multiscale finite element computations. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic integration. We validate this approach by introducing the new quadrature formulas into a multiscale finite element method for the two-dimensional reaction–diffusion equation.     

A piecewise approach to piecewise approximation

Tensor-product B-spline surfaces offer a convenient means for representing a set of bivariate data, especially if many surface evaluations are required. This is because the compact support property of the tensor-product spline allows the spline value to be obtained in a time that is (almost) independent of the number of coefficients used to define the surface. The main calculation is the precomputation involved in fitting the data and this can be impractically large if there are many spline coefficients to be calculated. Since the surface produced may be evaluated locally and efficiently, it would be advantageous to exploit local properties in order to fit the data in a piecewise manner. An algorithm to do this is presented.     

Global existence for the multi-dimensional compressible viscoelastic flows

The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces. Using uniform estimates for a hyperbolic-parabolic linear system with convection terms, we prove the global existence in the Besov space which is invariant with respect to the scaling of the associated equations. Several important estimates are achieved, including a smoothing effect on the velocity, and the L¹-decay of the density and deformation gradient.     

On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law

This paper is concerned with the existence and uniqueness of global, periodic and stationary solutions for flows of incompressible viscoelastic fluids for which the extra-stress tensor satisfies a differential constitutive law. More precisely, we prove that the results obtained by C Guillopé and J.C. Saut [5] remain true without any restriction on the smallness of the retardation parameter.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

A conservative pressure-correction scheme for transient simulations of reacting flows

An algorithm is presented for numerical simulations of time-dependent low Mach number variable density flows with an arbitrary amount of scalar transport equations and a complex equation of state. The pressure-correction type algorithm is based on a segregated solution formalism. It is conservative and guarantees stable results, regardless of the difference in density between neighboring cells. Furthermore, states are predicted which exactly match the equation of state. In the one-dimensional example, considering non-premixed flames, a simplified flamesheet model is used to describe the combustion of fuel and oxidizer. We demonstrate that the predicted states exactly correspond to the equation of state. We illustrate the accuracy improvement due to higher order formulation and demonstrate grid convergence.     

Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence

This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving L ²-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth.     

Forchheimer law derived by homogenization of gas flow in turbomachines

A general formulation of the homogenization problem of compressible fluid flow through a periodic porous material in turbomachines is presented here. This formulation is able to derive a Forchheimer law with a mean velocity dependent permeability as equivalent macroscopic behavior. To specify this permeability, additional flow problems are defined on the unit cell and solved by a mixed stabilized finite element discretization. The application of the Galerkin least-square (GLS) method requires the introduction of two stabilization terms with appropriate parameters. The mixed finite element discretization of these unit cell problems is finally outlined.     

A finite volume scheme for a model coupling free surface and pressurised flows in pipes

A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).