Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in R3 for any specific heat ratio γ 1.The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density,and the method of weak convergence.According to the author's knowledge,it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with γ 1.     

Two-sided a posteriori error bounds for electro-magnetostatic problems

This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between exact and approximate solutions of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not invoke specific properties of the approximation method. For this reason, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems. Bibliography: 24 titles.     

Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics

The dynamics of gaseous stars is often described by magnetic fields coupled to self-gravitation and radiation effects. In this paper we consider an initial-boundary value problem for nonlinear planar magnetohydrodynamics (MHD) in the case that the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes are taken into account. Based on the fundamental local existence results and global-in-time a priori estimates, we establish the global existence of a unique classical solution with large initial data to the initial-boundary value problem under quite general assumptions on the heat conductivity.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

Global Existence of a Weak Solution for a Model in Radiation Magnetohydrodynamics

We consider a simplified model based on the Navier-Stokes-Fourier system coupled to a transport equation and the Maxwell system, proposed to describe radiative flows in stars. We establish global-in-time existence for the associated initial-boundary value problem in the framework of weak solutions.     

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes’ equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by Maz’ya (DFG-Kolloquium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991) and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes’ equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström’s method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近[英文]

本文研究了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近问题，构造了定常Navier-Stokes方程对称破坏分歧点扩充系统及其谱Galerkin逼近扩充系统，证明了谱Galerkin逼扩充系统解的存在性和收敛性，从而给出了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近，并给出了逼近的误差估计。     

An approximation method using approximate approximations

The aim of this article is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present article we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In the first step, the unknown source density in the potential representation of the solution is replaced by approximate approximations. In the second, the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in the third, Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

Stability of non-constant steady-state solutions for bipolar non-isentropic Euler–Maxwell equations with damping terms

In this article, we consider the periodic problem for bipolar non-isentropic Euler–Maxwell equations with damping terms in plasmas. By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property. Furthermore, we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for bipolar non-isentropic Euler–Poisson equations. This phenomenon on the charge transport shows the essential relation and difference between the bipolar non-isentropic and the bipolar isentropic Euler–Maxwell/Poisson equations.     

Existence and stability of planar diffusion waves for 2-D Euler equations with damping

To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L₂ convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L₂ norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the Lp estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties.     

Nonlinear Schr?dinger Approximation for the Electron Euler-Poisson Equation

The nonlinear Schr?dinger(NLS for short) equation plays an important role in describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. In this paper, the authors study the NLS approximation by providing rigorous error estimates in Sobolev spaces for the electron Euler-Poisson equation,an important model to describe Langmuir waves in a plasma. They derive an approximate wave packet-like solution to the evolution equations by the multiscale a...     

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noether-type symmetry operators associated with partial Lagrangians. The ideas of the procedure for a system of unperturbed partial differential equations are extended to a system of perturbed or approximate partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. The theory is applied to the perturbed linear and nonlinear (1+1) wave equations and the Maxwellian tails equation. We have also obtained new approximate conservation laws for these equations.     

A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations

In this paper we are concerned with plane wave discretizations of nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. To this end, we design a plane wave method combined with local spectral elements for the discretization of such nonhomogeneous equations. This method contains two steps: we first solve a series of nonhomogeneous local problems on auxiliary smooth subdomains by the spectral element method, and then apply the plane wave method to the discretization of the resulting (locally homogeneous) residue problem on the global solution domain. We derive error estimates of the approximate solutions generated by this method. The numerical results show that the resulting approximate solutions possess high accuracy.     

A normal compliance contact problem in viscoelasticity: An a posteriori error analysis and computational experiments

In this work, the numerical approximation of a viscoelastic contact problem is studied. The classical Kelvin-Voigt constitutive law is employed, and contact is assumed with a deformable obstacle and modelled using the normal compliance condition. The variational formulation leads to a nonlinear parabolic variational equation. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize time derivatives. A priori error estimates recently proved for this problem are recalled. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and other parabolic equations. Upper and lower error bounds are proved. Finally, some numerical experiments are presented to demonstrate the accuracy and the numerical behaviour of the error estimates.     

Error estimates for approximate approximations with Gaussian kernels on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L₁-estimates are given, where all the constants are determined explicitly.     

An iterative method with error estimators

Iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions. In many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small. This paper presents a new iterative method based on the Lanczos process for the solution of linear systems of equations with a symmetric matrix. The method is designed to allow the computation of estimates of the Euclidean norm of the error in the computed approximate solutions. These estimates are determined by evaluating certain Gauss, anti-Gauss, or Gauss–Radau quadrature rules.     

Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier–Stokes (NSE) equations in fluid dynamics and Maxwell equations in eletromagnetism. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of each subprocess can require different meshes, time steps, and methods. In most terrestrial applications, MHD flows occur at low‐magnetic Reynold numbers. We introduce two partitioned methods to solve evolutionary MHD equations in such cases. The methods we study allow us at each time step to call NSE and Maxwell codes separately, each possibly optimized for the subproblem's respective physics. Complete error analysis and computational tests supporting the theory are given.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1083–1102, 2014     

Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier-Stokes equations in R³ for any specific heat ratio γ>1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.