Study of Three-Dimensional Boundary Value Problems of Fluid Filtration in an Anisotropic Porous Medium

Three-dimensional boundary value problems (the first and second boundary value problems and the conjugation problem) of stationary filtration of fluids in anisotropic (orthotropic) and inhomogeneous porous media are posed and studied. A medium is characterized by a symmetric permeability tensor whose components generally depend on the coordinates of points of the space. A nonsingular affine transformation of coordinates is used and the problems are stated in canonical form, which dramatically simplifies their study. In the case of orthotropic and piecewise orthotropic homogeneous medium, the solution of the problem with canonical boundaries (plane and ellipsoid surfaces) can be obtained in finite form. In the general case, where the orthotropic medium is inhomogeneous and the boundary surfaces are arbitrary and smooth, the problem can be reduced to singular and hypersingular integral equations. The problems are topical, for example, in the practice of fluid (water, oil) recovery from natural anisotropic and inhomogeneous soil strata.     

On a problem of viscous strip deformation with a free boundary

We consider a class of solutions to the 2D Navier-Stokes equations in a strip such that the longitudinal component of velocity is a linear function of the longitudinal coordinate while the transversal component and the pressure do not depend on this coordinate. One of the strip boundaries is free and the other boundary can be a solid wall or free too. We formulate conditions for a global solvability in time of corresponding initial boundary value problems, describe their asymptotic properties, give examples of exact solutions, study blowing up solutions in the case when the both strip boundaries are free.     

Dirichlet-Neumann alternating algorithm for an exterior anisotropic quasilinear elliptic problem

In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence. Moreover, the convergence rate is obtained in detail for a typical domain. Finally, some numerical examples are presented to illustrate the feasibility of the method.     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Composite mesh difference methods for elliptic boundary value problems

Summary The purpose of this paper is to develop composite mesh difference methods for elliptic boundary value problems over regions with curved, smooth boundaries. A curved mesh will cover an annular strip along the boundary of the region which is included in the mesh. For the rest of the region and for a suitable inner part of the annular strip a square or rectangular mesh will be used. On each mesh a difference approximation is set up as well as couplings between them. Only second order methods for second order elliptic equations will be treated in detail.This research was supported by the Swedish Institute for Applied Mathematics (ITM)     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,     

Estimates for deviations from exact solutions to plane problems in the cosserat theory of elasticity

A functional a posteriori estimate is obtained for control of the accuracy of approximate solutions to plane problems arising in the Cosserat theory of elasticity. We deal with the case where displacements and independent rotation are given on the boundary of the domain, a continuous medium is isotropic, and there is a linear dependence between stresses and strains. The proposed method is based on the duality theory of Calculus of Variations and can be also applied to the case of anisotropic media.     

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points lines, where three interfaces meet, and at the boundary points lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.     

The reducibility of the anisotropic Hele-Shaw problem to the isotropic case

It is established that the unilateral Hele-Shaw problem for flows in a channel when there is bulk anisotropy and Saffman–Taylor boundary conditions on the free boundary can be reduced to the isotropic case using a linear non-orthogonal coordinate transformation. Correspondingly, any exact solution of the Hele-Shaw problem for an isotropic medium generates a set of solutions for an anisotropic medium for arbitrary orientation of the principal axes of the permeability tensor with respect to the direction of the channel axis.     

Fehlerabschätzungen bei Randwertaufgaben partieller Differentialgleichungen mit unendlichem Grundgebiet

Summary Types of boundary value problems of partial differential equations for infinite domains are discussed which can easily be transformed in such a manner as to allow estimations of error (for approximate solutions) similar to the boundary maximum principle. First, second und third boundary value problems for the outer domain of linear elliptic and certain linear and nonlinear parabolic differential equations are examined. For elliptic differential equations one of the results is that the secound boundary value problem for more than two dimensions can be included. The estimates of the paper can thus be applied to problems of flow around some object, not in the case of two but of three dimensions. This is in a certain sense a counterpart to the conformal mappings method which is successful for two but not for three dimensions. Numerical examples show that estimations of error can easily be carried out.     

On the existence of weak solutions of anisotropic generally restrained plates

This paper presents investigations of free vibration of anisotropic plates of different geometrical shapes and generally restrained boundaries. The existence and uniqueness of weak solutions of boundary value problems and eigenvalue problems which correspond to the statical and dynamical behaviour of the mentioned plates is demonstrated. It is determined that when the plates have corner points formed by the intersection of edges free or elastically restrained against translation, the corresponding bilinear forms maintain the V – ellipticity property.     

Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In each problem, the value function is written as the solution of a free boundary problem involving second order ordinary differential equations, in which the unknown boundaries are found by applying the principle of smooth fit. For both problems, we compute the value function, we exhibit the optimal strategy and we prove its generic uniqueness.     

A nonlocal boundary value problem for elliptic differential-operator equations and applications

In this paper we give, for the first time, an abstract interpretation of nonlocal boundary value problems for elliptic differential equations of the second order. We prove coerciveness and Fredholmness of nonlocal boundary value problems for the second order elliptic differential-operator equations. We apply then, in section 6, these results for investigation of nonlocal boundary value problems for the second order elliptic differential equations (one can find the references on the subject in the introduction and Chapter V in the book by A. L. Skubachevskii [27]). Abstract results obtained in this paper can be used for study of nonlocal boundary value problems for quasielliptic differential equations.     

各向异性外问题的Schwarz 交替法及其收敛性和误差估计

本文对于无界区域各向异性常系数椭圆型偏微分方程研究了一种基于自然边界归化的Schwarz交替法.利用极值原理证明了在连续情形最大模意义下的几何迭代收敛性,通过选取适当的共焦椭圆边界利用Fourier分析获得了不依赖各向异性程度的最优的迭代收缩因子,还在离散情形最大模意义下证明了几何收敛性,而且进一步得到了误差估计,最后,数值结果证实了迭代收缩因子和误差估计的正确性,表明了该方法在无界Ⅸ域上求解各向异性椭圆型偏微分方程的优越性.     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Constructing orthogonal curvilinear meshes by solving initial value problems

Summary The purpose of this paper is to develop methods for constructing orthogonal curvilinear meshes suitable for solving partial differential equations over plane regions with smooth, curved boundaries. These curved meshes cover an annular strip along the boundary of the region which is included in the mesh. In this strip difference approximations of partial differential equations and boundary conditions can be set up as easily as they can for halfspace problems. The rest of the region and a suitable part of the annular strip can be covered by a square or rectangular mesh. In the present paper we consider the problem of determining curved meshes by solving nonlinear hyperbolic initial value problems which are formally related to the Cauchy-Riemann equations.This work was sponsored by the Swedish Institute for Applied Mathematics (ITM)     

Solution of plane seepage problems for a multivalued seepage law when there is a point source

The steady seepage of an incompressible fluid in a uniform porous medium, occupying an arbitrary bounded two-dimensional region, when there is a point source present is considered. Part of the boundary of the region is free, while the remaining part is impermeable for the fluid. It is assumed that the function defining the seepage law is multivalued and has a linear increase at infinity. A generalized formulation of the problem is proposed in the form of a variational inequality of the second kind. An approximate solution of the problem is obtained by an iterative splitting method, which enables approximate values of both the solution itself (the pressure) and its gradient to be found. Analytic expressions describing the boundaries of the region where the modulus of the pressure gradient takes a constant value are obtained for model problems of a line of bore holes. Numerical experiments are carried out for model problems, which confirm the effectiveness of the proposed method. Good agreement is observed between the results of calculations obtained analytically and by approximate methods.     

Numerical simulation and linear well-posedness analysis for a class of three-phase boundary motion problems

We investigate the linear well-posedness for a class of three-phase boundary motion problems and perform some numerical simulations. In a typical model, three-phase boundaries evolve under certain evolution laws with specified normal velocities. The boundaries meet at a triple junction with appropriate conditions applied. A system of partial differential equations and algebraic equations (PDAE) is proposed to describe the problems. With reasonable assumptions, all problems are shown to be well-posed if all three boundaries evolve under the same evolution law. For problems involving two or more evolution laws, we show the well-posedness case by case for some examples. Numerical simulations are performed for some examples.     

The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points

For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.