Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed‐point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.Copyright © 2014 John Wiley & Sons, Ltd.     

The fresh-salt water interface in a semi-pervious aquifer

The paper determines the location of the steady state interface between fresh and saltwater in a plane coastal aquifer. The lower boundary is totally impervious while the upper one is impervious below land and semi-pervious below the sea allowing an outflow through this part of the upper boundary. The model equations reduce to two boundary value problems, one valid in x < 0 and the other in x > 0 here x is measured along the aquifer with the origin at the coast. In each region unknown boundaries have to be determined as part of the solution using boundary and continuity conditions. Two cases are presented using the Dupuit approximation. One where the solution can be written down in terms of elementary functions and the other in which we have to use a phase space analysis.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

A system of boundary integral equations for polygonal plates with free edges

We consider the problem of a polygonal plate with free edges. It is a boundary value problem for the biharmonic operator on a polygon with Neumann boundary conditions. Its resolution is studied via boundary integral equations. A variational formulation of the boundary problem obtained by a double-layer potential is given. Finally, we implement the method and give numerical results. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Discrete absorbing boundary conditions for Schrödinger-type equations. Practical implementation

Recently, some absorbing boundary conditions for Schrödinger-type equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

     

Boundary integral equations for bending of thermoelastic plates with transmission boundary conditions

The solution of an initial‐boundary value problem for bending of a piecewise‐homogeneous thermoelastic plate with transverse shear deformation is represented as various combinations of single‐layer and double‐layer time‐dependent potentials. The unique solvability of the boundary integral equations generated by these representations is proved in spaces of distributions. Copyright © 2009 John Wiley & Sons, Ltd.     

Smoothness properties of Newtonian potentials in the study of elastic plates

The Hölder continuity and differentiability are investigated of Newtonian potentials arising in the theory of bending of elastic plates with transverse shear deformation. These properties play an essential role in the study by means of boundary integral equation techniques of boundary value problems for the equilibrium and harmonic oscillation states, and in the construction of associated boundary element methods.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

弹性力学问题解唯一的边界积分方程

从积分方程式出发,应用基本解的特性分析,说明在力边值问题中,位移边界积分方程和面力边界积分方程的位移解不唯一.提出了位移解唯一的条件,建立了唯一解的位移边界积分方程和面力边界积分方程.实例计算结果表明唯一解的边界积分方程是有效的.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary elements solution of stokes flow between curved surfaces with nonlinear slip boundary condition

This work presents a boundary integral equation formulation for Stokes nonlinear slip flows based on the normal and tangential projection of the Green's integral representational formulae for the velocity field. By imposing the surface tangential velocity discontinuity (slip velocity) in terms of the nonlinear slip flow boundary condition, a system of nonlinear boundary integral equations for the unknown normal and tangential components of the surface traction is obtained. The Boundary Element Method is used to solve the resulting system of integral equations using a direct Picard iteration scheme to deal with the resulting nonlinear terms. The formulation is used to study flows between curved rotating geometries: i.e., concentric and eccentric Couette flows and single rotor mixers, under nonlinear slip boundary conditions. The numerical results obtained for the concentric Couette flow is validated again a semianalytical solution of the problem, showing excellent agreements. The other two cases, eccentric Couette and single rotor mixers, are considered to study the effect of different nonlinear slip conditions in these flow configurations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 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.

     

Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g.around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two-dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl-Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.

     

Exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions

By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

求解对流扩散方程的一致四阶紧致格式

目前,许多高精度差分格式,由于未成功地构造与其精度匹配的稳定的边界格式,不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式,它的边界点的计算格式和内点的计算格式的截断误差主项保持一致,给出了具体内点和边界格式;并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度;在对流占优情况下,本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小,取得了不错的效果,理论结果得到了数值验证;驱动方腔数值结果显示,本文对N-S方程的离散格式具有很好的可靠性,适合对复杂流体流动的数值模拟和研究.     

Energetic boundary element method analysis of wave propagation in 2D multilayered media

The analysis of scalar wave propagation in 2D zonewise homogeneous media with vanishing initial and mixed boundary conditions is carried out. The problem is formulated in terms of time‐dependent boundary integral equations, and then it is set in a weak form, based on a natural energy identity satisfied by the differential problem solution. Several numerical results have been obtained by means of the related energetic Galerkin boundary element method showing accuracy and stability of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Semigroup Domination on a Riemannian Manifold with Boundary

We discuss the semigroup domination on a Riemannian manifold with boundary. Our main interest is the Hodge–Kodaira Laplacian for differential forms. We consider two kinds of boundary conditions; the absolutely boundary condition and the relative boundary condition. Our main tool is the square field operator. We also develop a general theory of semigroup commutation.     

Matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was previously known only in the second-order case and fourth-order case with separated boundary conditions.     

Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1-93] and Métivier's [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479] work. The linear stability results are based on Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95] work for the single shock front and Wang and Xin's [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529-558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479], we slightly modified Métivier's frame work to establish the existence for the solution to the nonlinear problem.