Free boundary problem for a two-layer inviscid incompressible fluid

The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.     

The discontinuous Galerkin method with reduced integration scheme for the boundary terms in almost incompressible linear elasticity

In this paper, the discontinuous Galerkin (dG) method is introduced and applied for a problem of nearly incompressible material behavior, where the standard finite element method, namely the conventional continuous Galerkin (cG) method faces the well-known problem of volumetric locking. The highlight of the work lies in the reduced integration scheme for the boundary terms of the dG method. Two different reduced and mixed integration schemes are presented and applied to reduce the calculation time. The dG method converges much faster than standard cG method with respect to the number of the elements, provided that the penalty value is sufficiently large. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust boundary element method for nearly incompressible linear elasticity

Summary. For a Dirichlet boundary value problem in linear elasticity we consider a boundary element method which is robust for nearly incompressible materials. Based on the spectral properties of the single layer potential for the Stokes problem we introduce an orthogonal splitting of the trial space. The resulting variational problem is then well conditioned and can be discretized by using standard boundary element methods. Mathematics Subject Classification (1991):65N38     

Solvability of a problem on the plane motion of a viscous incompressible fluid with a free noncompact boundary

One considers the problem of the plane motion of a viscous incompressible fluid which fills partially a container V, bounded by the straight line ₁ = {x:x ₂ = 0} and the contour (V₁), consisting of two semilines ⁽¹⁾ = {x:x ₁<–1,x ₂ = h₀} ⁽²⁾ = {x:x ₁ = 0,x ₂h₀+1} joined by a smooth curvel ⁽³⁾. One assumes that the motion is due to a nonzero flow and by the motion of the lower wall ₁ with a constant velocity R0. The unique solvability of this problem is proved for small R and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 174–179, 1981.In conclusion, the author expresses his deep gratitude to V. A. Solonnikov for his guidance.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

On the initial‐boundary value problem of the incompressible viscoelastic fluid system

In this paper, we shall establish the local well‐posedness of the initial‐boundary value problem of the viscoelastic fluid system of the Oldroyd model. We shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state. © 2007 Wiley Periodicals, Inc.     

Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid

The authors construct the trajectory attractor and global attractor for an autonomous two-dimensional non-Newtonian fluid.     

Smooth solutions for the motion of a ball in an incompressible perfect fluid

In this paper we investigate the motion of a rigid ball surrounded by an incompressible perfect fluid occupying RN. We prove the existence, uniqueness, and persistence of the regularity for the solutions of this fluid-structure interaction problem.     

Analytical integration and exact geometrical representation in the two-dimensional elastostatic boundary element method

Two ways of improving the accuracy of results in the boundary element method are considered. Since the geometries of many problems of practical interest are created from straight lines and circular arcs, errors caused by representing such geometries approximately using quadratic shape functions can be removed using exact geometrical representations for straight and circular boundaries. Besides, exact geometrical representations enable exact analytical integrations for some situations, thereby eliminating the errors caused by approximate numerical integration. The results of some simple test problems show that the use of exact representation of straight and circular geometries, and analytical integration in the situations where this is possible, offers worthwhile benefits in the boundary element analysis of two-dimensional elastostatics problems.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

On the numerical integration of a boundary value problem involving a fourth order linear differential equation

A finite difference method for the numerical integration of the linear two-point boundary value problem $$y^{(4)} + f(x)y = g(x),y(a) = A_1 ,y(b) = A_2 ,y''(a) = B_1 ,y''(b) = B_2$$ is constructed and analyzed. The method usesf′,f″,g′,g″ at the boundary points to obtain anO(h ⁶) global error, while requiring only the solution of a system of linear equations associated with a five-band matrix. A typical numerical example is included to demonstrate the practical usefulness of the numerical procedure as well.     

A domain decomposition and mixed method for a linear parabolic boundary value problem

In this paper, we discuss the convergence of a domain decompositionmethod for the solution of linear parabolic equations in theirmixed formulations. The subdomain meshes need not be quasi-uniform;they are composed of triangles or quadrilaterals that do notmatch at interfaces. For the ease of computation, this lackof continuity is compensated by a mortar technique based onpiecewise constant (discontinuous) multipliers. It is shownthat the method on triangles, parallelograms or slightly distortedparallelograms is convergent at the expense of a half-orderloss of accuracy compared with mortar methods based on piecewiselinear multipliers.     

A variational method applied to a linear hyperbolic initial boundary value problem

This paper deals with the application of a variational method to a boundary value problem of the wave equation. Starting with an initial boundary value problem (which is given) introduction of a boundary condition at the final time leads to a boundary value problem with one of the initial conditions redundant. This redundant initial condition is used by the trial function of the direct method (of the Ritz type) which is employed to stationarize the variational principle.     

Random attractor for a two-dimensional incompressible non-Newtonian fluid with multiplicative noise

This article proves that the random dynamical system generated by a twodimensional incompressible non-Newtonian fluid with multiplicative noise has a global random attractor, which is a random compact set absorbing any bounded nonrandom subset of the phase space.     

Preconditioning for boundary control problems in incompressible fluid dynamics

PDE‐constrained optimization problems arise in many physical applications, prominently in incompressible fluid dynamics. In recent research, efficient solvers for optimization problems governed by the Stokes and Navier–Stokes equations have been developed, which are mostly designed for distributed control. Our work closes a gap by showing the effectiveness of an appropriately modified preconditioner to the case of Stokes boundary control. We also discuss the applicability of an analogous preconditioner for Navier–Stokes boundary control and provide some numerical results.     

Dynamics of a rigid body in a two-dimensional incompressible perfect fluid and the zero-radius limit

Science China Mathematics - In this survey we report some recent results on the dynamics of a rigid body immersed in a two-dimensional incompressible perfect fluid, with an emphasis on the...     

Flows of an incompressible fluid with linear vorticity

All the flows of an incompressible fluid, for which the vorticity of the velocity field is linear with respect to the spatial variables, are described.