Regularity and perturbation results for mixed second order elliptic problems

We study a mixed boundary value problem for elliptic second order equations obtaining optimal regularity results under weak assumptions on the data. We also consider the dependence of the solution with respect to perturbations of the boundary sets carrying the Dirichlet and the Neumann conditions.     

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.     

Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations

We consider the solvability of boundary value problems and some inverse problems for operator-differential equations of odd order which can be referred to as mixed type equations since the coefficient of the higher order time derivative can change sign. Studying the general classes of boundary value problems with nonlocal boundary conditions, we establish the existence and uniqueness theorems of regular solutions under some conditions on data.

     

Solvability of a boundary value problem for operator-differential equations of mixed type

We study boundary value problems for operator-differential equations of mixed type, prove existence of generalized solutions, and establish their smoothness in weighted Sobolev spaces. The results are applied to odd order forward-backward equations.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

A System of Differential-Operator Equations of Different Orders in Hilbert Spaces

We consider an abstract Cauchy problem for a system of nonhomogeneous abstract differential equations in Hilbert spaces. The “main” equation is of the second order and “boundary” equations are of the first order. Existence of a solution is proved. Application to mixed (initial boundary-value) problems for one-dimensional second order hyperbolic equations and for fourth order PDEs with the time derivative in boundary conditions has been shown. The first author was partially supported by 60% funds of the University of Bologna and G.N.A.M.P.A. of INdAM; the second author was supported by the Israel Ministry of Absorption.     

Non-coercive Linear Elliptic Problems

We study here some linear elliptic partial differential equations (with Dirichlet, Fourier or mixed boundary conditions), to which convection terms (first order perturbations) are added that entail the loss of the classical coercivity property. We prove the existence, uniqueness and regularity results for the solutions to these problems.     

Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes

The solution of the three-dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A ‘tensor-product’ decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence of the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence.     

A fourth‐order compact algorithm for nonlinear reaction‐diffusion equations with Neumann boundary conditions

In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction‐diffusion equations is fourth‐order accurate in both the temporal and spatial dimensions. We also prove that the standard second‐order approximation to zero Neumann boundary conditions provides fourth‐order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Boundary control problem for the first boundary value problem for a second-order system of hyperbolic type

We consider a boundary control problem for a system of second-order hyperbolic equations without the mixed derivative. The boundary functions are constructed. We state a theorem that gives existence conditions for boundary controls.     

Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition

Summary We consider a mixed finite element approximation of the stationary, incompressible Navier-Stokes equations with slip boundary condition, which plays an important rôle in the simulation of flows with free surfaces and incompressible viscous flows at high angles of attack and high Reynold's numbers. The central point is a saddle-point formulation of the boundary conditions which avoids the well-known Babuka paradox when approximating smooth domains by polyhedrons. We prove that for the new formulation one can use any stable mixed finite element for the Navier-Stokes equations with no-slip boundary condition provided suitable bubble functions on the boundary are added to the velocity space. We obtain optimal error estimates under minimal regularity assumptions for the solution of the continous problem. The techniques apply as well to the more general Navier boundary condition.     

Immersed finite element methods for 4th order differential equations

We propose three new finite element methods for solving boundary value problems of 4th order differential equations with discontinuous coefficients. Typical differential equations modeling the small transverse displacement of a beam and a thin plate formed by multiple uniform materials are considered. One important feature of these finite element methods is that their meshes can be independent of the interface between different materials. Finite element spaces based on both the conforming and mixed formulations are presented. Numerical examples are given to illustrate capabilities of these methods.     

Integral boundary value problems for first order integro-differential equations with deviating arguments

This paper deals with first order integro-differential equations of mixed type with deviating arguments. We investigate the existence of solutions of such problems with integral boundary conditions by establishing a comparison result and applying the monotone iterative technique. To obtain corresponding results, we also discuss first order differential inequalities with deviating arguments. Two examples demonstrate the application of our results.     

Stable second-order finite-difference methods for linear initial-boundary-value problems

Finite-difference methods of second order at the boundary points are presented for problems with one-dimensional second-order hyperbolic and parabolic equations with mixed boundary conditions. These methods do not require information at points outside the region of consideration. The linear stability of the algorithms developed is investigated. Numerical experiments are given for illustrating the accuracy and stability of the methods. Though the focus is on homogeneous boundary conditions, finite-difference methods with non-homogeneous mixed boundary conditions are also developed. To show the potential of the methods developed, in terms of CPU time, a comparison is made with the Crank–Nicolson method.     

高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.     

拟线性混合方程的不连续边值问题

该文利用复分析的方法对双曲区域实施共形映照,通过将双曲区域的边界曲线变换为直线段,研究了另一类更一般区域上的拟线性二阶混合型方程的不连续边值问题,得到了一些较新的可解性结果的推广.     

Mixed boundary element solution for three-dimensional convection-diffusion problem with a velocity profile

A mixed boundary element formulation is presented for convection-diffusion problems with a velocity profile. In this formulation the convection-diffusion equation is considered as a nonlinear diffusion equation with inhomogeneous terms in which the convective term is involved additionally, because the spatial distribution of the drift velocity cannot be straightforwardly expressed in boundary integral form. Accordingly, a corresponding boundary integral equation may be described usually in the form of a so-called hybrid-type boundary integral equation.

In the present paper, mixed boundary elements are employed in a discrete model of the original convection-diffusion system. In the mixed element, potentials are approximated linearly, and their normal derivatives to boundaries are assumed constant. A simple iterative scheme is adopted in order to solve hybrid-type mixed boundary element equations. Simple three-dimensional models are dealt with in numerical experiments. The proposed approach gives more accurate and stable solutions compared with constant boundary elements which have been reported.