Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

Steklov-Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions

We study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions. We introduce the notion of “eigenvalue-lines” in the plane; these eigenvalue-lines join each Steklov eigenvalue to the first eigenvalue of the Neumann problem with homogeneous boundary condition. We prove existence results when the nonlinearities involved asymptotically stay, in some sense, below the first eigenvalue-lines or in a quadrilateral region (depicted in Fig. 1) enclosed by two consecutive eigenvalue-lines. As a special case we derive the so-called nonresonance results below the first Steklov eigenvalue as well as between two consecutive Steklov eigenvalues. The case in which the eigenvalue-lines join each Neumann eigenvalue to the first Steklov eigenvalue is also considered. Our method of proof is variational and relies mainly on minimax methods in critical point theory.     

Patterns in parabolic problems with nonlinear boundary conditions

We obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator in such domains. This information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and a uniform trace theorem.     

Discrete Sturm-Liouville problems with nonlinear parameter in the boundary conditions

This paper deals with discrete second order Sturm-Liouville problems where the parameter that is part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions. By constructing the Poincare operator, we obtain the existence of -periodic weak solutions under some reasonable assumptions.     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Existence of positive solutions for singular ordinary differential equations with nonlinear boundary conditions

We prove the existence of nonnegative solutions of the problem , , for a physically motivated class of nonlinearity . The results, which are established using a ``forbidden value' argument, are new even in the case of linear .

     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions

We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups.     

A BOUNDARY ELEMENT METHOD FOR A NONLINEAR BOUNDARY VALUE PROBLEM IN STEADY-STATE HEAT TRANSFER IN DIMENSION THREE

In this paper, a new method of boundary reduction is proposed, which reduces thesteady-state heat transfer equation with radiation. Moreover, a boundary element method is pre-sented for its solution and the error estimates of the numerical approximations are given.     

Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions

Let Ω⊂RN be a bounded domain with Lipschitz boundary, with a>0 on . Let σ be the restriction to ∂Ω of the (N−1)-dimensional Hausdorff measure and let be σ-measurable in the first variable and assume that for σ-a.e. x∈∂Ω, B(x,⋅) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p∈(1,∞), the operator Ap:=div(a|∇u|^p−2∇u) with nonlinear Wentzell-Robin type boundary conditions

     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

一类具非线性边值条件的非线性方程的奇摄动问题

研究了一类具非线性边值条件的非线性方程的奇摄动问题,运用合成展开法构造了问题的形式渐近解,并用微分不等式理论证明了所得渐近解的一致有效性.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions

In this paper we propose a new modified recursion scheme for the resolution of boundary value problems (BVPs) for second-order nonlinear ordinary differential equations with Robin boundary conditions by the Adomian decomposition method (ADM). Our modified recursion scheme does not incorporate any undetermined coefficients. We also develop the multistage ADM for BVPs encompassing more general boundary conditions, including Neumann boundary conditions.     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions

We study nth order boundary value problems with a nonlinear term f(t,x) subject to nonhomogeneous multi-point boundary conditions. Criteria for the existence of positive solutions of such problems are established. Conditions are determined by the relationship between the behavior of f(t,x)/x near 0 and ∞ when compared with the smallest positive characteristic value of an associated linear integral operator. This work improves and extends some recent results in the literature for the second order problems. The results are illustrated with examples.