Convergence of the solution for a domain decomposed,heterogeneously modeled flow past a concave profile problem

The free streamline problem investigated is that of fluid flow past a symmetric truncated concave‐shaped profile between walls. An open wake or cavity is formed behind the profile. Conformal mapping techniques are used to solve this problem. The problem formulated in the hodograph plane is decomposed into two nonoverlapping domains. Heterogeneous modeling is then used to describe the problems, i.e., a different governing differential equation in each domain. In one of these domains, a Baiocchi‐type transformation is used to obtain a fixed domain formulation for the part of the transformed problem containing an unknown boundary. In the other domain, the Baiocchi‐type transformation is extended across the boundary between the two domains, thus yielding a different problem formulation. This also assures that the dependent variables and their normal derivatives are continuous along this common boundary. The numerical solution scheme, a successive over‐relaxation approach, is applied over the whole problem domain with the use of a projection‐operation over only the fixed domain formulated part. Numerical results are obtained for the case of a truncated circular profile. These results are found to be in good agreement with another published result. The existence and uniqueness of the solution to the problem as a variational inequality is shown, and the convergence of the numerical solution using a domain decomposition method scheme is demonstrated by assuming some convergence property on the common interface of the two subdomains. © 2000 John Wiley & Sons, Inc. Numeer Methods Partial Differential Eq 16: 459–479, 2000     

Alternating iteration and elliptic variational inequalities

Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the free boundary part of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given.     

Optimal Shape Problems for Eigenvalues

ABSTRACT

We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.     

Macro-Hybrid Variational Formulations of Constrained Boundary Value Problems

In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping domain decompositions, and corresponding three-field versions are derived. Then, for regularization purposes, augmented formulations are established via preconditioned exact penalizations and expressed in terms of proximation operators. Optimization interpretations are given for potential problems, recovering the classic two- and three-field augmented Lagrangian formulations. Furthermore, associated parallel two- and three-field proximal-point algorithms are discussed for numerical resolution of finite element discretizations. Applications to dual mixed variational formulations of problems from mechanics illustrate the theory.     

Variational procedures for a class of exterior interface problems

The problem under consideration is that of determining a function which is a solution of the Helmholtz equation in a planar region exterior to a simple closed curve and of an inhomogeneous Helmholtz equation inside the curve. Jump conditions on the function and its normal derivative across the cruve are given. The problem is first transformed into one involving the inner region only with a boundary condition which is non-local. This means that the solution at a point on the boundary is a functional of its values elsewhere. This second problem is further transformed into a variational form with all boundary conditions natural. It is shown that the variational problem has a solution. Finite dimensional approximate problems are defined and they are shown to have solutions converging to the solution of the variational problem.     

Analysis of augmented three-field macro-hybrid mixed finite element schemes

On the basis of composition duality principles, augmented three-field macro- hybrid mixed variational problems and finite element schemes are analyzed. The compatibility condition adopted here, for compositional dualization, is the coupling operator surjectivity, property that expresses in a general operator sense the Ladyenskaja-Babuka-Brezzi inf-sup condition. Variational macro-hybridization is performed under the assumption of decomposable primal and dual spaces relative to nonoverlapping domain decomp...     

Global stress regularity of convex and some nonconvex variational problems

We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains. These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a difference quotient technique. In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems. The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case).      

凹角区域双曲外问题的精确人工边界条件

研究一类凹角区域双曲型外问题的数值方法.先用Newmark方法对时间进行离散化,在每个时间步求解一个椭圆外问题.然后引入人工边界,并获得精确的人工边界条件.给出半离散化问题的变分问题,证明了变分问题的适定性,并给出了误差估计.最后给出数值例子,以示该方法的可行性与有效性.     

Higher Critical Points in an Elliptic Free Boundary Problem

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.     

广义弹塑性梁理论及接触问题中的时偶变分不等式*

为研究摩擦接触问题,本文建立了一个具有二类独立交量的二维弹塑性梁模型。由此提出了一个新的非线性二次互补性问题。其中的外部互补性条件定义了自由边界;而内部互补性条件则控制了弹塑性分界面。文中证明了此二次互补性问题等价于一非线性变分不等式,并导出了其对偶变分不等式。本文结果显示对偶问题较原问题有更多的优越性。应用于塑性极限分析理论中,文中最后证明了一个简单的下限定理。     

解单障碍问题的Lagrangian乘子非匹配网格区域分解法

针对二阶椭圆型单障碍问题提出了一类基于非匹配网格的Lagrang ian乘子非重叠型区域分解方法.并在适当条件下给出了该方法的收敛性分析和收敛速度估计.     

ON THE COUPLING OF BOUNDARY INTEGRAL AND FINITE ELEMENT METHODS FOR SIGNORINI PROBLEMS

1.IntroductionPartialdifferentialequationssubjecttounilateralboundaryconditionsareusuallycalledSignoriniproblemsintheliterature.TheseproblemshavebeenstudiedbymanyauthodssincetheappearenceofthehistoricalpaperbyA.Signoriniin1933[25].Signoriniproblemsaroseinmanyareasofapplicationse.g.,theelasticitywithunilateralconditions[lo],thefluidmechnicsproblemsinmediawithsemipermeableboundaries[8,12],theelectropaintprocess[1]etc.Fortheexistence,uniquenessandregularityresultsforSignorinitypeproblemswerefer…     

Free boundary problems with nonlinear source terms

Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.This research was supported by the U.S. Army Research Office under Contract DAAG-79-0145     

Variational formulations of nonlinear boundary-value problems with a free boundary in the theory of interaction of surface waves with acoustic fields

Variational problems equivalent to nonlinear evolutionary boundary-value problems with a free boundary are formulated. These problems arise in the theory of interaction of limited volumes of liquid, gas, and their interface with acoustic fields. It is proved that the principle of separation of motions can be applied to these variational problems. The problem of a capillary-acoustic equilibrium form is given in a variational formulation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1642–1652, December, 1993.     

On some oblique derivative problems arising in the fluid flow in porous media. A theoretical and numerical approach

In this paper we consider some free boundary problems related to the fluid flow in a porous medium. By applying a method due to Baiocchi [1] these problems are reduced to nonlinear problems on a fixed domain. The main difficulty here lies in the fact that such problems are not variational because of jump discontinuities in the direction of the oblique derivative in the boundary condition. We give a uniqueness result and by a constructive method we establish at the same time an existence result and a new algorithm for the numerical solution of the original free boundary problem. Some numerical results are given.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

Parabolic variational inequality with parameter and gradient constraints

This paper concerns a singular control problem whose value function is governed by a time-dependent HJB equation with gradient constraints. The method is to transform a two-dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies. Moreover we analyze the behaviors of the free boundary surfaces. The main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter.     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Variational problems with obstacles and integral constraints

Various problems in mathematics and physics can be formulated in terms of a variational problem with obstacles and integral constraints, e.g. finding a surface of minimal area with prescribed volume in a bounded region.We are concerned with the regularity of solutions of variational problems: We show that the minima of a variational integral under all Sobolewfunctions with prescribed boundary values, lying between two obstacles, and fulfilling some integral constraints, are bounded and Hölder-continuous. We do not assume any differentiability or convexity of the integrand, but only a Caratheodory and a growth condition.This research has been supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft.     

Front propagation in infinite cylinders. II. The sharp reaction zone limit

This paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction–diffusion–advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples.