The convergence of non-Newtonian fluids to Navier-Stokes equations

In this paper, the convergence of solutions for incompressible dipolar viscous non-Newtonian fluids is investigated. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equations in the sense of L²-norm (resp. H¹-norm), as the viscosities tend to zero and the initial data belong to H¹(Ω) (resp. H²(Ω)). Moreover, we obtain L^∞-norm convergence of solutions if the initial data belong to H²(Ω).     

Global behavior of spherically symmetric Navier-Stokes equations with density-dependent viscosity

In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in L^∞-norm and weighted H¹-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.     

$H^2$-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this...     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

An error estimate uniform in time for spectral semi-galerkin approximations of the nonhomogeneous navier-stokes equations

We consider the spectral semi-Galerkin method applied to the non-homogeneous Navier-Stokes equations, which describes the motion of miscibles fluids. Under certain conditions it is known that the aproximate solutions constructed by using this method converge to a global strong solution of these equations. In this paper we prove that these solutions satisfy an optimal uniform in time error estimate in the H ¹-norm for the velocity. We also derive an uniform error estimate in the L ^∞-norm for the density and an improved error estimate in the L ²-norm for the velocity.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Controllability of neutral stochastic evolution equations driven by fractional brownian motion

In this paper,we investigate the controllability for neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter H ∈(1/2,1) in a Hilbert space.We employ the α-norm in order to reflect the relationship between H and the fractional power α.Sufficient conditions are established by using stochastic analysis theory and operator theory.An example is provided to illustrate the effectiveness of the proposed result.     

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain(ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations.Precisely,for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme,and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero,then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time.The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper.Furthermore,we prove that,in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws,the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws.This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms.Experimental results which confirm the theoretical results are presented.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.     

A coupling method based on new MFE and FE for fourth-order parabolic equation

In this article, a coupling method of new mixed finite element (MFE) and finite element (FE) is proposed and analyzed for fourth-order parabolic partial differential equation. First, the fourth-order parabolic equation is split into the coupled system of second-order equations. Then, an equation is solved by finite element method, the other equation is approximated by the new mixed finite element method, whose flux belongs to the square integrable space replacing the classical H(div;Ω) space. The stability for fully discrete scheme is derived, and both semi-discrete and fully discrete error estimates are obtained. Moreover, the optimal a priori error estimates in L ² and H ¹-norm for both the scalar unknown u and the diffusion term γ and a priori error estimate in (L ²)²-norm for its flux σ are derived. Finally, some numerical results are provided to validate our theoretical analysis.     

High accuracy analysis of the lowest order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H¹-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h²)/O(h² + τ²) in H¹-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.     

Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes

We derive a nonlinear stabilized Galerkin approximation of the Laplace operator for which we prove a discrete maximum principle on arbitrary meshes and for arbitrary space dimension without resorting to the well-known acute condition or generalizations thereof. We also prove the existence of a discrete solution and discuss the extension of the scheme to convection–diffusion–reaction equations. Finally, we present examples showing that the new scheme cures local minima produced by the standard Galerkin approach while maintaining first-order accuracy in the H¹-norm. To cite this article: E. Burman, A. Ern, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

EQ 1 rot nonconforming finite element method for nonlinear dual phase lagging heat conduction equations

EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.     

A class of nonconforming quadrilateral finite elements for incompressible flow

This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow.Beyond the previous research works,we propose a general strategy to construct the basis functions.Under several specific constraints,the optimal error estimates are obtained,i.e.,the first order accuracy of the velocities in H1-norm and the pressure in L2-norm,as well as the second order accuracy of the velocities in L2-norm.Besides,we clarify the differences between rectangular and quadrilateral finite element approximation.In addition,we give several examples to verify the validity of our error estimates.     

Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R³. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H³-framework. Moreover, if additionally the initial data belong to L^p with , the optimal convergence rates of the solutions in L^q-norm with 2≤q≤6 and its spatial derivatives in L²-norm are obtained.     

A symmetric finite volume scheme for selfadjoint elliptic problems

Based on a linear finite element space, a symmetric finite volume scheme for a self-adjoint elliptic boundary-value problem is proposed. Error estimates in L²-norm, H¹-norm, and L^∞-norm are derived. Some post-processing techniques are also provided.     

Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L^∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L^∞-norm.     

Stabilized finite element method for the viscoelastic Oldroyd fluid flows

In this paper, a new stabilized finite element method based on two local Gauss integrations is considered for the two-dimensional viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newtonian fluid flows. This new stabilized method presents attractive features such as being parameter-free, or being defined for non-edge-based data structures. It confirms that the lowest equal-order P ₁???P ₁ triangle element and Q ₁???Q ₁ quadrilateral element are compatible. Moreover, the long time stabilities and error estimates for the velocity in H ¹-norm and for the pressure in L ²-norm are obtained. Finally, some numerical experiments are performed, which show that the new method is applied to this model successfully and can save lots of computational cost compared with the standard ones.