Finite element derivative interpolation recovery technique and superconvergence

A new finite element derivative recovery technique is proposed by using the polynomial interpolation method. We show that the recovered derivatives possess superconvergence on the recovery domain and ultraconvergence at the interior mesh points for finite element approximations to elliptic boundary problems. Compared with the well-known Z-Z patch recovery technique, the advantage of our method is that it gives an explicit recovery formula and possesses the ultraconvergence for the odd-order finite elements. Finally, some numerical examples are presented to illustrate the theoretical analysis.     

Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid

We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L ₂-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.     

A Calculus of Boundary Value Problems in Domains with Non–Lipschitz Singula Points

The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.     

位势问题边界元法中几乎奇异积分的正则化

将一种通用算法应用于平面位势问题边界元法中近边界点几乎奇异积分的正则化。对线性单元,位势问题近边界点的几乎强和超奇异积分可归纳为两种形式。通过分部积分,将引起奇异的积分元素变换到积分号之外,从而对这两种积分分别给出了无奇异的正则化计算公式。除了线性元,二次元也应用于该算法。与近边界点临近的二次单元划分为两段线性单元,该算法仍然适用。算例证明了方法的有效性和精确性。对曲线边界问题,联合二次元和线性元可提高计算结果精确度。     

Stable Ergodicity and Frame Flows

In this note we show that all diffeomorphisms close enough to the time-one map of the frame flow on certain negatively curved manifolds are ergodic. As a simple corollary we deduce that the frame flows are ergodic for all compact manifolds with curvature pinched sufficiently close to –1, thus providing results in the case of manifolds of dimension 7 or 8 which were missing from the results of Brin and Karcher.     

A bounce back-immersed boundary-lattice Boltzmann model for curved boundary

In this paper, a bounce back-immersed boundary-lattice Boltzmann model(BB-IB-LBM) is proposed for curved boundary.In the present model, a modified density distribution function is proposed for curved boundary including stationary and moving curved boundaries. A special treatment is also developed to satisfy no-slip boundary condition for the curved boundary with large curvature. On Lagrangian boundary points, the modified distribution functions are implemented to replace the artificial correction force in conventional immersed boundary-lattice Boltzmann methods(IB-LBMs). Numerical experiments are given to illustrate the accuracy and efficiency of present BB-IB-LBM. The drag coefficient of the test cases predicted by the present model is in better agreement with the results of experimental results than that of the previous IB-LBMs. It is also concluded that the average drag coefficient of present model are consistent with the experimental results. Comparing with conventional IB-LBMs, the present model eliminates the non-physical vortex at the tail of an airfoil. Simulation of flow over a sphere also proves the extensibility of present method in three-dimensional simulation.     

Complex variable equations and the numerical solution of harmonic problems for piecewise-homogeneous media

Complex variable boundary integral equations are derived using of holomorphicity theorems for plane harmonic problems concerning unit structures with inclusions, pores and lines of discontinuity of the potential and/or the flow. Unlike the method of analytical elements, the equations cover problems in which discontinuities in the potential, flow and conductance can simultaneously be encountered at the contact points. Versions of the equations are given for connected half planes and for periodic and biperiodic problems. Formulae are obtained which determine the effective impedance tensor of the equivalent homogeneous medium in cases when the unit structure is biperiodic or when the representative volume of a structured medium is identified with the basic cell of a biperiodic system. Recurrence quadrature formulae are proposed which enable one to solve the resulting equations effectively using the complex variable boundary element method. They indicate the computational advantages of using the complex variable method compared with the real variable method: the three integrals appearing in the resulting equations are evaluated analytically in the case of linear elements (regular and singular) with the densities approximated using algebraic polynomials of arbitrary degree. In the case of elements (regular and singular) in the form of an arc of a circle, only one integral requires numerical integration when the densities are approximated using complex trigonometrical polynomials of arbitrary degree. It is emphasized that the combination of the linear and curved boundary elements which have been developed enables the smooth part of a contour to be approximated while retaining the continuity of the tangent and avoiding the complications which arise when the smoothness of the approximation of a contour is ensured using conformal mapping. Examples are presented which illustrate the computational merits of the method developed. They show a sharp increase in accuracy (by orders of magnitude) when curved elements are used for the curvilinear parts of a contour and when terminal elements are used to calculate the flow intensity coefficient at singular points (the crack tips the vertices of angular notches and the common vertices of the units of the medium).     

Contact problems of the theory of elasticity in the presence of wear : PMM vol.40, n 6, 1976, pp. 981–986

The solutions of the contact problems of the theory of elasticity in the presence of wear is given for two cases. In Sect. 1 we consider the problems in which an initially curved beam comes in contact with a half-plane. One of the initial assumptions is that the distance between certain directrices along which the body in contact is sliding and the boundary of the half-plane remains constant. In Sect. 1 the contact between the curved beam and the half-plane is discussed at the assumption that the half-plane is subject to wear. As the result of the wear, the pressure between the beam and the half-plane is gradually reduced. It is naturally assumed that the pressure at the terminal points of the contact area will, in this case, be zero. The conditions characterizing the pressure at these terminal points can be established for various types of contact problems only under certain additional assumptions; this will be discussed below.     

The Wiener disorder problem with finite horizon

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation.     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

An Algorithm of Linear Combinations: Thermal Conduction

This paper presents computational algorithms that make it possible to overcome some difficulties in the numerical solving boundary value problems of thermal conduction when the solution domain has a complex form or the boundary conditions differ from the standard ones. The boundary contours are assumed to be broken lines (the 2D case) or triangles (the 3D case). The boundary conditions and calculation results are presented as discrete functions whose values or averaged values are given at the geometric centers of the boundary elements. The boundary conditions can be imposed on the heat flows through the boundary elements as well as on the temperature, a linear combination of the temperature and the heat flow intensity both at the boundary of the solution domain and inside it. The solution to the boundary value problem is presented in the form of a linear combination of fundamental solutions of the Laplace equation and their partial derivatives, as well as any solutions of these equations that are regular in the solution domain, and the values of functions which can be calculated at the points of the boundary of the solution domain and at its internal points. If a solution included in the linear combination has a singularity at a boundary element, its average value over this boundary element is considered.     

The Use of Parabolic Arcs in Matching Curved Boundaries in the Finite Element Method

If isoparametric coordinates are used to deal with curved boundariesin the finite element method, the original boundary is implicitlyreplaced by a series of parabolic or cubic arcs. The equationsof these arcs involve parameters which are the coordinates ofpoints on the curved side, and a simple procedure is outlinedfor choosing these parameters in such a way that each arc isa parabola which passes through four points of the originalcurve thus ensuring a good approximation to it.     

Characteristic Boundary Layers for Mixed Hyperbolic-Parabolic Systems in One Space Dimension and Applications to the Navier-Stokes and MHD Equations

We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so-called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second-order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null electrical resistivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems. © 2020 Wiley Periodicals, Inc.     

On global solutions of gravity driven nonhomogeneous viscous flows in a bounded domain

We consider an initial boundary value problem for nonhomogeneous Navier‐Stokes equations with a uniform gravitational field. For any given steady density profile whose derivatives are sufficiently close to a negative constant, we show that there exists a unique global solution if the initial perturbation with respect to the steady state is sufficiently small.     

Integral equations for the Fourier-transformed boundary values for the transmission problems for right-angled wedges and octants

A 4×4-system of integral equations for the Fourier transformed boundary values of the normal derivatives of the wave functions defined in the four quadrants of R ²-space is derived. This system results from the scalar transmission problem with continuous passage of the boundary values of the total wave-fields and of the weighted normal derivatives corresponding to the case of magnetically polarized fields. Several equivalent systems of integral equations are deduced then which show that Banach's fixed point principle may be applied at least for slightly differing media in the four quadrants. The method, which is equivalent to a compatibility condition for holomorphic functions, may be generalized to the case of the scalar transmission problem for octants in R ³-space. There a 12 × 12-system of integral equations for the Fourier transformed normal derivatives on the quarter-plane faces is established.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

DARBOUX EQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITH NONNEGATIVE CURVATURE IN R^3

The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate on the boundary is contained. As some applications the smooth isometric embeddings of positively and nonnegatively curved disks into Rs are constructed.     

Inverse scattering for the magnetic Schrödinger operator

We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n?3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston (1995) [5] which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schrödinger equation via a pseudodifferential conjugation argument.     

Modified combined grid method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A modified combined grid method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. The six-point averaging operator is applied at next-to-the-boundary grid points, while the 18-point averaging operator is used instead of the 26-point one at the remaining grid points. Assuming that the boundary values given on the faces have fourth derivatives satisfying the Hölder condition, the boundary values on the edges are continuous, and their second derivatives obey a matching condition implied by the Laplace equation, the grid solution is proved to converge uniformly with the fourth order with respect to the mesh size.