A uniform error estimate in time for spectral Galerkin approximations of the magneto‐micropolar fluid equations

We consider Galerkin approximations for the equations modeling the motion of an incompressible magneto‐micropolar fluid in a bounded domain. We derive an optimal uniform in time error bound in the H¹ and L² ‐norms for the velocity. This is done without explicit assumption of exponential stability for a class of solutions corresponding to decaying external force fields. Our study is done for no‐slip boundary conditions, but the results obtained are easily extended to the case of periodic boundary conditions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 689–706, 2012     

A note on uniform in time error estimates for approximations to reaction-diffusion equations

The approximation of solutions of reaction-diffusion equationsthat approach asymptotically stable, hyperbolic equilibria isconsidered. Near such equilibria trajectories of the equationcontract and hence it is possible to seek error estimates thatare uniformly valid in time. A technique for the derivationof such estimates is illustrated in the context of an explicitEuler finite-difference scheme.     

An error estimate for matrix equations

This paper proposes a new method for estimating the error in the solution of matrix equations. The estimate is based on the adjoint method in combination with small sample statistical theory. It can be implemented simply and is inexpensive to compute. Numerical examples are presented which illustrate the power and effectiveness of the new method.     

Hermite pseudospectral approximations. An error estimate

We find an error bound for the pseudospectral approximation of a function in terms of Hermite functions, hn(x)=e−x²Hn(x), in certain weighted Sobolev spaces. We use properties of Hermite polynomials, as well as an asymptotic expression for their largest zeroes to achieve the mentioned estimate.     

An error estimate for Volterra integral equations

A bound is given for the error incurred by using Gregory's integration formula for solving nonlinear Volterra equations of the second kind.     

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Summary. This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the -norm in the situation, where the diffusion parameter is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results. Received February 25, 1999 / Revised version received July 6, 1999 / Published online August 2, 2000     

Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low‐frequency and high‐frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov‐type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ?², and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments T^ε of size O(ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second‐order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u^ε(t) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.     

A uniform estimate for the ELLAM scheme for transport equations

We prove a priori error estimate in a weighted energy norm for the Eulerian‐Lagrangian localized adjoint method (ELLAM) for the transport equations, without any special refinement or numerical stabilization introduced. The estimate holds uniformly with respect to ?. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary. In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law with initial condition . The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is in space-time -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case. Received October 21, 1999 / Published online February 5, 2001     

Continuous dependence on the time and spatial geometry for the navier-stokes equations

In this paper we derive explicit a priori inequalities which imply stability of the solution of the initial-boundary value problem for the Navier-Stokes equations under perturbations of the initial time geometry and of the spatial geometry. These inequalities bound the solution perturbation In L₂ in terms of some well defined measure of the perturbation in geometry. We establish continuous dependence on spatial geometry in both two and three dimensions and continuous dependence on initial geometry in two dimensions. In the latter problem the three dimensional case will be somewhat more complicated due to the different form of the Sobolev inequality.     

广义Benjamin-Bona-Mahony方程Fourier谱方法的大时间误差估计

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｅｎｊａｍｉｎ Ｂｏｎａ Ｍａｈｏｎｙｅｑｕａｔｉｏｎｕｔ+ｕｘ+ｕｕｘ -ｕｘｘ-ｕｘｘｔ =0 ( 1 .1 )ｉｎｃｏｒｐｏｒａｔｅｓｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅａｎｄｄｉｓｓｉｐａｔｉｖｅｅｆｆｅｃｔｓ ,ａｎｄｈａｓｂｅｅｎｐｒｏｐｏｓｅｄａｓａｍｏｄｅｌｆｏｒｂｏｔｈｔｈｅｂｏｒｅｐｒｏｐａｇａｔｉｏｎａｎｄｔｈｅｗａｔｅｒｗａｖｅｓ[1,2 ] .Ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｉｓｅｑｕａｔｉｏｎｈａｖｅｂｅｅ…     

An error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes

We give here an error estimate for a finite volume discretization of the Stokes equations in two space dimensions on equilateral triangular meshes. This work was initiated by an analogous result presented by Alami‐Idrissi and Atounti for general triangular meshes. However, in this latter article, the result is not actually proven. We state here the restricting assumptions (namely equilateral triangles) under which the error estimate holds, using the tools which were introduced by Eymard, Gallouet and Herbin and used by Alami‐Idrissi and Atounti. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

An oscillation criterion for nonhomogeneous half-linear differential equations

By means of a classical inequality and an averaging technique, we obtain an oscillation criterion for nonhomogeneous half-linear differential equations. Our result is much more general than a recent result of Wong [1] and makes use of the oscillatory behavior of the forcing terms on intervals.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations

A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.     

On partial regularity results for the navier-stokes equations

Looking at the regularity results of Scheffer, respectively, Caffarelli, Kohn and Nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the Navier-Stokes equations.     

An estimate of the error of the linearized equations of motion of an axisymmetric projectile

An estimate is obtained, by small-parameter methods, of the error of the solution for the system of equations of motion of a projectile in the atmosphere, linearized for small angles of attack with respect to the variables describing the angular oscillations of the axis of symmetry, compared with the solution of the initial non-linear system for the same initial conditions.