Stability and convergence of semi-implicit time-stepping algorithm for stationary incompressible magnetohydrodynamics

For the stationary incompressible magnetohydrodynamics (MHD) equations, we provide a new uniqueness assumption (A0) and show the exponential stability of the solution. Then, the semi-implicit time-stepping algorithm is used to solve the stationary MHD equations. The algorithm is proved to be unconditionally stable. The discrete velocity and magnetic field are bounded in $L^{\infty }(0,+\infty ;{\mathit{L}}^{\infty }\left(\Omega \right))$ for any space and time step sizes. The error estimates for the algorithm are deduced under the uniqueness conditions. Finally, numerical experiments are performed to testify our theoretical analysis.     

A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

We introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem, in two and three dimensions. The velocity field is discretized using divergence-conforming Brezzi–Douglas–Marini (BDM) elements and the magnetic field is approximated by curl-conforming Nédélec elements. The H¹-continuity of the velocity field is enforced by a DG approach. A central feature of the method is that it produces exactly divergence-free velocity approximations, and captures the strongest magnetic singularities. We prove that the energy norm error is convergent in the mesh size in general Lipschitz polyhedra under minimal regularity assumptions, and derive nearly optimal a priori error estimates for the two-dimensional case. We present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensional problems.     

Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions

This paper investigates the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order α∈(1,2]. Our results are based on some standard fixed point theorems. Some examples are presented to illustrate the main results.     

Numerical approximation of one-dimensional stationary diffusion equations with boundary layers

This article is aimed at showing, in a very simple physical context, how one can improve the numerical solution of singular problems involving boundary layers. Taking into account information on the structure of the boundary layer can result in significant simplification and increase of accuracy. Finite elements methods are considered in this article, other discretization methods and more involved problems including higher dimensional problems and the time dependant case will be discussed in Methods and Applications of Analysis (in press), and elsewhere. Besides the numerical aspects, it is noteworthy that the approach to boundary layer expansion, presented in the Appendix, does not require matched asymptotic expansion techniques.     

具Cauchy-Ventcel边界的阻尼波方程的对数衰减性

本文研究有界区域上具Cauchy-Ventcel边界条件的波动方程的解的衰减性质。在不要求耗散区域满足几何控制条件的情形下,我们得到了波方程的对数衰减结果。 主要结果的证明依赖于具Cauchy-Ventcel边界条件的椭圆方程的插值不等式以及关于该椭圆方程的预解式估计。为得到期望的插值不等式, 我们采用的工具是整体Carleman估计。     

Elastic buckling of rectangular mindlin plate with mixed boundary conditions

An approximate method for analyzing the elastic buckling of Mindlin plates is proposed. The solutions of the differential equations of buckling of Mindlin plate are obtained in discrete forms by applying numerical integration. The discrete solutions give the transverse shear forces, twisting moment, bending moments, rotations and deflection at the all discrete points which are the intersections of the vertical and horizontal equally dividing lines on the plate. In order to confirm the convergency and accuracy of numerical solutions, the buckling loads of simply supported square Mindlin plates are calculated, and the results are compared with other published finite element solutions. As an application, the buckling loads and buckling modes of Mindlin plates with various types of mixed boundaries are calculated.     

Large amplitude vibrations of circular plates with mixed boundary conditions

In this paper, a method of analysing large amplitude vibrations of circular plates with mixed boundary conditions is explained and is illustrated with an example where part of the boundary is clamped and the remaining simply-supported     

Multiple solutions for fractional differential equations with nonlinear boundary conditions

In this paper, we study certain fractional differential equations with nonlinear boundary conditions. By means of the Amann theorem and the method of upper and lower solutions, some new results on the multiple solutions are obtained.     

Integration of second order linear differential equation with mixed boundary conditions

A fourth order finite difference scheme for obtaining an approximate solution of second order linear differential equation lacking the first derivative, with mixed boundary conditions, is presented. The convergence of the method is proved. A numerical illustration is included to demonstrate the practical usefulness of our method.     

Fixed sign solutions of second-order difference equations with Neumann boundary conditions

In this paper, we study the existence of solution of the nonlinear second-order difference problem with Neumann boundary conditions. Assuming the existence of a pair of ordered lower and upper solutions γ and β, we obtain optimal existence results for the case γ ≤ β and even for γ ≥ β.To this end, we do an exhaustive study of the values of the real parameters α and μ, for which the associated Green's function to the linear operator L[α,μ] u_κ ≡ u_κ+2 − 2 α u_κ+1 + μ u_κ with Neumann boundary conditions has fixed sign.     

Finite-difference solution of the incompressible three-dimensional boundary layer equations for a blunt body

The governing equations for a laminar flow are solved in terms of an orthogonal surface coordinate system. One of the coordinate is determined by the intersection with the body surface of meridional planes which pass through an axis containing the stagnation point. The other coordinate is obtained numerically from the orthogonality condition. The momentum equations have been written in a standard from which allows additional equations of this form to be added with a small modification of the computer code. This equation is replaced with a nonlinear finite-difference equation which is solved as an iterative solution of linear tridiagonal equations. The special form of the governing equations at the stagnation point and the plane of symmetry is determined and the solution of these equations is obtained to provide a unified code. Numerical solutions have been obtained for several special cases and compared to results of other authors. New results are presented for an ellipsoid ar angle of attack and an elliptic-paraboloid at zero incidence.     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.     

Generalized Euler-Lagrange equations for fractional variational problems with free boundary conditions

In this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems (FVPs) with completely free boundary conditions have an extremum. The fractional derivatives are defined in the Caputo sense. First we present the necessary conditions for the problem with only one dependent variable, and then we generalize them to problems with multiple dependent variables. We also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables. It is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives, the new results reduce to the known necessary conditions. Some examples are presented to demonstrate the applicability of the new formulations.     

Direct methods of solution of partial differential equations with periodic boundary conditions

In this paper, suitable discrete approximations for the general elliptic and parabolic partial differential equation with periodic boundary conditions are derived and appropriate direct and fast solution methods of the resulting linear systems proposed.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

热弹耦合梁方程组在非线性边界条件下的整体吸引子

本文研究了一类在非线性边界条件下的热弹耦合梁方程组的初边值问题,首先通过先验估计证明系统存在唯一的整体解,其次通过证明系统存在有界吸收集和半群的渐近光滑性得到整体吸引子的存在性.     

Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques

We consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary. As a paradigm for such problems, we simply refer to Stokes flow. After a brief review of the problem and of its well posedness, we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints, we exploit a penalty method to account for them. We perform the analysis of the method in terms of consistency, boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization. Since the additional terms introduced to account for the defective boundary conditions are non-local, we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy. In contrast to alternative discretization methods based on Lagrange multipliers accounting for the constraints on the boundary, the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation. Indeed, it does not require neither multipliers nor sub-iterations or additional adjoint problems with respect to the reference problem at hand.     

第一类边界对流扩散方程LDG方法的稳定性

针对一维常系数对流扩散模型方程,讨论了当含有第一类边界条件时,局部间断有限元方法（LDG方法）的稳定性。利用有限元理论基本分析技巧,证明了当边界条件为第一类的边界条件时,LDG方法为稳定的,并利用数值算例证明理论分析的正确性。