Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$-framework

We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the $L^r$-setup. The mathematical model used here is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated using three types of boundary conditions: the conditions of periodicity on the “lower” and “upper” parts of the boundary, the Dirichlet boundary conditions on the “inflow” and on the profile and an artificial “do nothing”-type boundary condition on the “outflow.” Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in ${\mathbf{W}}^{1,r}(\mathrm{\Omega })$ and its continuous dependence on the data. We explain the sense in which the “do nothing” boundary condition on the “outflow” is satisfied.     

Numerical conformal mapping of multiply connected domains to regions with circular boundaries

We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.     

Minimization principles for elliptic hemivariational inequalities

In this paper, we explore conditions under which certain elliptic hemivariational inequalities permit equivalent minimization principles. It is shown that for an elliptic variational–hemivariational inequality, under the usual assumptions that guarantee the solution existence and uniqueness, if an additional condition is satisfied, the solution of the variational–hemivariational inequality is also the minimizer of a corresponding energy functional. Then, two variants of the equivalence result are given, that are more convenient to use for applications in contact mechanics and in numerical analysis of the variational–hemivariational inequality. When the convex terms are dropped, the results on the elliptic variational–hemivariational inequalities are reduced to that on “pure” elliptic hemivariational inequalities. Finally, two representative examples from contact mechanics are discussed to illustrate application of the theoretical results.     

Solvability and stability of von Karman model without rotational inertia with nonlinear forces

We investigate the nonlinear oscillations of plates described by von Karman evolution equations with mixed–hinged/simply supported nonlinear boundary conditions. The von Karman evolution system is of “hyperbolic type” with nonlinear transverse and boundary forces. We prove the existence of global solution (on given interval) constructing new variational approach to that system. We do not prove the Hadamard well-posedness, however a kind of new dependence of solutions on initial data is considered.     

Solutions to the seepage face model for dual porosity flows with hysteresis

This paper is concerned with theoretical analysis of the initial-boundary value problem for the dual porosity model of Darcian flows with the Preisach hysteresis operator. Due to the presence of the so-called unilateral boundary condition, the problem comes as an evolution variational inequality. We apply the penalty method to prove the global-in-time existence of the variational solution by constructing approximate solutions satisfying suitable energy type estimates.     

Signorini接触问题的边界变分不等式

本以Signorini接触问题为背景，讨论了变分不等式与边值问题的等价性，利用Green公式，基本解和基本解法向导数的性质，将区域型变分不等式化成等价的边界型变分不等式，并证明了边界变分不等式解的存在唯一性，为使用边界元方法数值求解提供理论依据。     

Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

In this paper we study the geometric numerical solution of the so called “good” Boussinesq equation. This goal is achieved by using a convenient space semi‐discretization, able to preserve the corresponding Hamiltonian structure, then using energy‐conserving Runge–Kutta methods in the Hamiltonian boundary value method class for the time integration. Numerical tests are reported, confirming the effectiveness of the proposed method.     

A preconditioner for generalized saddle point problems: Application to 3D stationary Navier‐Stokes equations

In this article we consider the stationary Navier‐Stokes system discretized by finite element methods which do not satisfy the inf‐sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen‐type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices (“generalized saddle point problems”). We show that if the underlying finite element spaces satisfy a generalized inf‐sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1‐P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier‐Stokes flows confirm our results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Two-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case

We introduce an approximation procedure and provide existence results for two-phase flow equations in porous media. The medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations. Our main interest is the outflow boundary condition which models an interface with open space. The approximate system introduces standard boundary conditions and can be used in numerical schemes. It allows the derivation of maximum principles. This is the basis for the derivation of the limiting system in the form of a variational inequality.     

The Elastic–Plastic Torsion Problem: A Posteriori Error Estimates for Approximate Solutions

We consider the variational inequality that describes the torsion problem for a long elasto-plastic bar. Using duality methods of the variational calculus, we derive a posteriori estimates of functional type that provide computable and guaranteed upper bounds of the energy norm of the difference between the exact solution and any function from the corresponding energy space that satisfies the Dirichlet boundary condition.     

On a boundary variational inequality of the second kind modelling a friction problem

A scalar contact problem with friction governed by the Yukawa equation is reduced to a boundary variational inequality. The presence of the non‐differentiable friction functional causes some difficulties when approximated. We present two methods to overcome this difficulty. The first one is a regularization leading to a non‐linear boundary variational equation, for which we propose an iterative procedure, whereas the second method is based on the boundary mixed variational formulation involving Lagrange multipliers. We propose Uzawa's algorithm to compute the saddle point of the corresponding boundary Lagrangian and investigate the discretization of various formulations by the boundary element Galerkin method. Convergence of the boundary element solution is proved and a convergence order is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

摩擦问题中的边界混合变分不等式

本文以弹性力学中的摩擦问题为背景,讨论了非线性、不可微的混合变分不等式解的存在唯一性,给出相应的边界变分不等式及其解的存在唯一性。为使用边界元方法数值求解提供了理论依据。     

One time‐step finite element discretization of the equation of motion of two‐fluid flows

We discretize in space the equations obtained at each time step when discretizing in time a Navier‐Stokes system modelling the two‐dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions at the boundary and at the interface between the two fluids. We discretize this system with the “mini‐element” and establish error estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The Allen–Cahn equation with dynamic boundary conditions and mass constraints

The Allen–Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint, which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as a variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well‐posedness result is proved for the related initial value problem. Copyright © 2014 John Wiley & Sons, Ltd.     

On the boundary element method for the Signorini problem of the Laplacian

Summary For the Laplace equation with Signorini boundary conditions two equivalent boundary variational inequality formulations are deduced. We investigate the discretization by a boundary element Galerkin method and obtain quasi-optimal asymptotic error estimates in the underlying Sobolev spaces. An algorithm based on the decomposition-coordination method is used to solve the discretized problems. Numerical examples confirm the predicted rate of convergence.     

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

We consider a simple reaction-diffusion system exhibiting Turing’s diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.     

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.     

Nontopological N‐vortex condensates for the self‐dual Chern‐Simons theory

We prove the existence of nontopological N‐vortex solutions for an arbitrary number N of vortex points for the self‐dual Chern‐Simons‐Higgs theory with 't Hooft “periodic” boundary conditions. We use a shadowing‐type lemma to glue together any number of single vortices obtained as a perturbation of a radially symmetric entire solution of the Liouville equation. © 2003 Wiley Periodicals, Inc.     

Dynamic thermoviscoelastic problem with friction and damage

In this paper we present a model of dynamic frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a temperature effect described by the parabolic equation with the subdifferential boundary condition and a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition. Contact is modeled with bilateral condition and is associated to a subdifferential frictional law. The variational formulation of the problem leads to a system of hyperbolic hemivariational inequality for the displacement, parabolic hemivariational inequality for the temperature and parabolic variational inequality for the damage. The existence of a unique weak solution is proved by using recent results from the theory of hemivariational inequalities, variational inequalities, and a fixed point argument.