HIGH-ORDER-ACCURATE DISCRETIZATION STENCIL FOR AN ELLIPTIC EQUATION

The coefficients for a nine-point high-order-accurate discretization scheme for an elliptic equation ∇²u− γ²u=r₀ (∇² is the two-dimensional Laplacian operator) are derived. Examples with Dirichlet and Neumann boundary condtions are considered. In order to demonstrate the high-order accuracy of the method, numerical results are compared with exact solutions.     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Convergence Rates in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> for Elliptic Homogenization Problems

We study rates of convergence of solutions in L ² and H ^1/2 for a family of elliptic systems {L_e}{\{\mathcal{L}_\varepsilon\}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_e}{\{\mathcal{L}_\varepsilon\}} . Most of our results, which rely on the recently established uniform estimates for the L ² Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains.     

Concentration of Solutions for Some Singularly Perturbed Mixed Problems: Existence Results

In this paper,we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.     

Mixed Boundary Value Problems for the Stationary Navier–Stokes System in Polyhedral Domains

The authors consider boundary value problems for the Navier–Stokes system in a polyhedral domain, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are arbitrarily combined on the faces of the polyhedron. They prove existence and regularity theorems for weak solutions in weighted (and nonweighted) L _p Sobolev and Hölder spaces with sharp integrability and smoothness parameters.     

The Viscosity Method for the Homogenization of Soft Inclusions

In this paper, we consider periodic soft inclusions T _ε with periodicity ε, where the solution, u _ε , satisfies semi-linear elliptic equations of non-divergence in \({\Omega_{\epsilon}=\Omega\setminus \overline{T}_\epsilon}\) with Neumann data on \({\partial T^{\mathfrak a}}\). The difficulty lies in the non-divergence structure of the operator where the standard energy method, which is based on the divergence theorem, cannot be applied. The main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of u _ε , referred to as u, as ε approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of a second corrector is then developed to show that the limit, u, is the viscosity solution of a homogenized equation.     

Scattering of scalar waves from a rough interface using a single integral equation

Using Green's function methods we consider the problem of scattering from a rough interface separating two semi-infinite homogenous media. We derive a single coordinate-space integral equation of the first kind for the generalized reflection coefficient R. A second integral equation of the first kind is derived for the generalized transmission coefficient T. The two equations are new results. In the limiting cases corresponding to Dirichlet and Neumann boundary conditions, we recover the usual boundary integral equations from the R-equation. In the flat surface limit, R and T reduce to the usual Fresnel reflection and transmission coefficients. The latter are derived from these more general results rather than assumed as in perturbation methods.     

Blow-up of solutions of nonlinear pseudo-hyperbolic equations of generalized nerve conduction type

This paper deals with the two types of mixed problems with respect to Neumann boundary and Dirichlet boundary for nonlinear pseudo-hyperbolic equations of generalized nerve conduction type when the nonlinear part F(x, t, u, u, u _t)and the initial values satisfy some conditions, the blow-up properties of the solutions are obtained.     

Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}\kern-1pt D^{\alpha}u = f(t,u)$ , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(??{0})) and lim _x→0 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}\kern-1pt D$ is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.     

Entropy Solutions for Nonlinear Degenerate Problems

We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)_t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.     

Reconstruction Formula for Identifying Cracks

We consider an inverse boundary value problem for identifying cracks in a conductive medium. By combining the probe method and an analysis for the behavior of the “reflected solution”, we derive a reconstruction formula for identifying cracks from the Neumann to Dirichlet map. We give also some related results.     

On the existence of solutions of boundary value problems of duffing type systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ [1 ] ,ｕｎｄｅｒａｇｒｏｕｐｏｆｖｅｒｙｅｘｔｅｎｓｉｖｅｃｏｎｄｉｔｉｏｎｓ,ＳＨＥＮＺｕ＿ｈｅｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｕｎｉｑｕｅ 2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆｔｈｅｓｙｓｔｅｍｏｆｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｕ″(ｔ) Ｇ(ｕ(ｔ) ) =ｐ(ｔ) ,(1 )ｗｈｅｒｅＧ :Ｒｎ →Ｒｈａｓａｃｏｎｔｉｎｕｏｕｓｓｅｃｏｎｄｐａｒｔｉａｌｄｅｒｉｖａｔｉｖｅｓ,ａｎｄｐ:Ｒ→Ｒｎｉｓｃｏｎｔｉｎｕｏｕｓａｎｄ2π＿ｐｅｒ…     

Regularity Issues in the Problem of Fluid Structure Interaction

We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier–Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have Hölder regularity C ^1,α , 0 < α ≦ 1. First, we show the existence and uniqueness of strong solutions up to the collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard H ² estimate for smoother domains. Then, we study the asymptotic behaviour of one C ^1,α body falling over a flat surface. We show that a collision is possible in finite time if and only if α < 1/2.     

On regular and singular perturbations of acoustic and quantum waveguidesSur des perturbations régulières et singulières dans des guides d'ondes acoustiques et quantiques

We consider regular and singular perturbations of the Dirichlet and Neumann boundary value problems for the Helmholtz equation in n-dimensional cylinders. The existence of eigenvalues and their asymptotics are studied. To cite this article: R.R. Gadyl'shin, C. R. Mecanique 332 (2004).     

Prise en compte des effets non linéaires de surface libre en écoulement instationnaire

This Note describes a computational method for three dimensional unsteady flows around a submerged body with forward speed. The two free-surface boundary conditions are written under their non linear form. The calculations are carried out in the time domain using a mixed Euler–Lagrange scheme based on the knowledge, at each time step, of the potential on the free surface and of the location of this surface. A mixed problem with a Neumann condition on the body and a Dirichlet one on the free surface is then solved. The panel method uses desingularized sources to represent free surface effects. Validations are carried out on steady flows. To cite this article: A. Rebeyrotte et al., C. R. Mecanique 333 (2005).     

Effet de la rugosité sur un fluide laminaire avec conditions de Fourier

The effect of tiny asperities covering a wall on a flow governed by Stokes equations with Fourier boundary conditions is investigated. We calculate the limit flow and we give estimates of the deviations of the drag, velocity field and pressure, in terms of the size ε of the asperities. In the particular case of a plate, the limit drag is larger than the drag of the smooth wall, in contrast with the situation found for Dirichlet boundary conditions.     

A Hopf bifurcation with Robin boundary conditions

We investigate Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations. By performing a smooth homotopy of boundary conditions from Neumann to Dirichlet type, we observe the creation of branches of periodic solutions with submaximal symmetry in codimension two bifurcations, although we do not fully calculate the branching behaviour. We also note that mode interactions behave generically on varying the boundary conditions. The investigation is performed using a numerical Liapunov-Schmidt reduction technique of Ashwin, Böhmer, and Mei (1994) and an analysis of Swift (1988).     

Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

This paper is devoted to constructing a general theory of nonnegative solutions for the equation called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ?⁺ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > m _c = max {(N? 2)/N,0}, in which the limits of classical solutions are also continuous in ? ^N as extended functions with values in ?₊∪{∞}. We introduce a precise class of extended continuous solutions ? _c and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ? _c has an initial trace in ?⁺, and (iii) that the solutions in ? _c are limits of classical solutions. Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈??, t>0. Well-posedness is established for this problem for m _c < m > 1 when ? is any open subset of ? ^N and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=f ^q posed in any open set ?. For 1 > q > N/(N? 2)₊ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense. As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦m _c , where the limits of smooth solutions need not be extended-continuously.     

New numerical method for partial differential equations. 1: Application to the diffusion equation

In this paper a new, highly accurate method called PH is presented for the numerical integration of partial differential equations. The method is applied for the solution of the one-dimensional diffusion equation. Upon integrating the equation within a subdomain of space and time using the prismoidal approximation, a three-point implicit scheme is obtained with a truncation error of order O(k⁴, h⁶), where k and h represent the time and space steps respectively. The method is stable under the condition s = αk/h² ? S(δ), where the function S(δ) increases as the parameter δ decreases from 1/12 to negative values. In practice the method behaves as unconditionally stable upon choosing an appropriate value for δ. A new formula is also adopted for the implementation of a Neumann boundary condition, introducing a truncation error of order O(h⁴). Numerical solutions are obtained incorporating Dirichlet and Neumann boundary conditions. The results prove that our method is far more accurate than any other-implicit or explicit method.