Existence of solutions to the nonhomogeneous steady Magnetohydrodynamic equations

We study the nonhomogeneous boundary value problem for the steady Magnetohydrodynamic equations in a two-dimensional bounded domain with multiply connected boundary. We prove that this problem has an admissible solution in an admissible domain if the boundary value is admissible. The proof of the main result uses some property for a weak solution to the transport equations in an admissible domain.     

Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions

An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.

     

NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.     

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Asymptotics of eigenvalues of the Dirichlet boundary value problem for the Lame operator in a three-dimensional domain with a small cavity

A boundary value problem for the Lame operator in a bounded three-dimensional domain with a small cavity is studied. The domain is filled with an elastic homogeneous isotropic medium that is clamped at the boundary, which corresponds to the Dirichlet boundary condition. The leading term of an asymptotic expansion for the eigenvalue is constructed in the case of the Dirichlet limit problem. The asymptotic expansion is constructed in powers of a small parameter ? that is the diameter of the cavity.     

On Mappings of Plane Domains by Solutions of Second-Order Elliptic Equations

In this paper, we study sufficient conditions for the one-to-one solvability of secondorder partial differential equations in a plane Jordan domain. For a continuous one-to-one and orientation-keeping map of the boundary of a Jordan domain to the rectifiable boundary of some other Jordan domain, we prove the following property: If the Cauchy integral whose measure is generated by this map is bounded by some constant in the exterior domain, then the solution to the corresponding Dirichlet problem in the domain with this boundary function maps these domains one-to-one. In the proof of the main result we use integral representations of equation solutions, particularly, properties of Fredholm-type integral equations on the domain boundary.     

Absorbing boundary conditions for electromagnetic wave propagation

In this paper, the theoretical perfectly absorbing boundary condition on the boundary of a half-space domain is developed for the Maxwell system by considering the system as a whole instead of considering each component of the electromagnetic fields individually. This boundary condition allows any wave motion generated within the domain to pass through the boundary of the domain without generating any reflections back into the interior. By approximating this theoretical boundary condition a class of local absorbing boundary conditions for the Maxwell system can be constructed. Well-posedness in the sense of Kreiss of the Maxwell system with each of these local absorbing boundary conditions is established, and the reflection coefficients are computed as a plane wave strikes the artificial boundary. Numerical experiments are also provided to show the performance of these local absorbing boundary conditions

     

Numerical methods for determining the inhomogeneity boundary in a boundary value problem for Laplace’s equation in a piecewise homogeneous medium

A boundary value problem for Laplace’s equation in a bounded two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary and the solution of the equation given the solution and its normal derivative on the boundary of the domain is discussed. Numerical methods are proposed for solving the inverse problem, and the results of numerical experiments are presented.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

The trace to the boundary of Sobolev spaces on a snowflake

The trace to the boundary is defined for functions in a Sobolev space in a domain with fractal boundary, for instance von Koch's snowflake domain. The image and the kernel of the trace operator are characterized. Partially supported by the Swedish Natural Science Research Council     

The Point Spectrum of the Linearized Boltzmann Operator with the External-force Term in a Bounded Domain

We will investigate the point spectrum on the imaginary axis of the linearized Boltzmann operator with an external-force potential in a bounded domain whose boundary is sufficiently smooth. The boundary condition considered is the perfectly reflective boundary condition. The point spectrum on the imaginary axis is only equal to {0}. However, the null space varies with the common axes of symmetry of the domain and the external-force potential.     

A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations

Summary. We analyze the error of a fictitious-domain method with boundary Lagrange multiplier. It is applied to solve a non-homogeneous steady incompressible Navier-Stokes problem in a domain with a multiply-connected boundary. The interior mesh in the fictitious domain and the boundary mesh are independent, up to a mesh-length ratio. Received February 24, 1999 / Revised version received January 30, 2000 / Published online October 16, 2000     

Iterative method for solving a three-dimensional electrical impedance tomography problem in the case of piecewise constant conductivity and one measurement on the boundary

The problem of electrical impedance tomography in a bounded three-dimensional domain with a piecewise constant electrical conductivity is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem is to determine the surface that is the boundary of the inhomogeneity from given measurements of the potential and its normal derivative on the outer boundary of the domain. An iterative method for solving the inverse problem is proposed, and numerical results are presented.     

Condensation of fermion pairs in a domain

We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two-body potential which admits a bound state. The particles are confined to a domain with Dirichlet boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross–Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

无界区域上Stokes问题的自然边界元与有限元耦合法

§1.引言 对于用有限元方法求解平面有界区域上的Stokes问题,国内外已有大量工作,例如可见[2]、[9]及其所引文献.但对无界区域上的这一问题,由于区域的无界性给有限元方法带来了困难,边界元方法及边界元与有限元的耦合法便显示其优越性.本文提出用自然边界元与有限元的耦合法求解无界区域上的Stokes问题.这一耦合法早在作者以前的工作中被应用于求解调和问题、重调和问题和平面弹性问题,但将它用于求解     

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.     

Necessary and sufficient conditions for unique determination of plane domains

We say that a domain U ? ?ⁿ is uniquely determined from the relative metric of its Hausdorff boundary (the relative metric is the extension by continuity of the intrinsic metric of the domain to the boundary) if every domain V ? ?ⁿ with the Hausdorff boundary isometric in the relative metric to the Hausdorff boundary of U is isometric to U too (in the Euclidean metrics). In this article we state some necessary and sufficient conditions for a plane domain to be uniquely determined from the relative metric of its Hausdorff boundary.     

Artificial boundary conditions for elliptic systems on polyhedral truncation surfaces

For a sufficiently broad class of formally selfadjoint boundary value problems in the domains with conical outlets to infinity, including exterior boundary value problems, we suggest an algorithm for constructing some artificial boundary conditions on polyhedral truncation surfaces that guarantees higher precision of approximation as the infinite domain is replaced with a large but bounded domain. The errors are estimated. Anisotropic 3-dimensional elasticity problems are discussed as an example.