Neumann Dirichlet Research Articles

A D-N alternating algorithm for exterior 3-D problem with ellipsoidal artificial boundary

<abstract><p>In this study, based on a general ellipsoidal artificial boundary, we present a Dirichlet-Neumann (D-N) alternating algorithm for exterior three dimensional (3-D) Poisson problem. By using the series concerning the ellipsoidal harmonic functions, the exact artificial boundary condition is derived. The convergence analysis and the error estimation are carried out for the proposed algorithm. Finally, some numerical examples are given to show the effectiveness of this method.</p></abstract>

Numerical investigation of the two– dimensional time–dependent diffusion equation using Radial basis functions

In this paper, we investigate the numerical solution of the two–dimensional time–dependent diffusion equation with non-local and mixed Neumann–Dirichlet boundary conditions. In the discretization process, the backward Euler as well as Crank–Nicolson schemes and radial basis function (RBF) collocation method are respectively used to discretize time derivative and spatial derivative terms. The accuracy and applicability of the presented methods are illustrated and compared by solving two examples.

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann–Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $$H^1$$ norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the $$H^1$$ -type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.

Spatial Patterns and Bifurcation Analysis of a Diffusive Tumour-immune Model

In this paper, a diffusive tumour-immune model is presented. By comparing the effect of Neumann boundary conditions and Dirichlet boundary conditions on the stability of trivial equilibrium, we derive that the former can provide more mechanisms for spatial pattern formation of the model. By taking the diffusion rate of tumour cells as a parameter, we first give the local and global steady-state bifurcations emitting from the positive equilibrium of the model. Then the stability of the bifurcation solution is discussed by computing the second derivative of an appropriate function, which is different from the general case. Furthermore, numerical simulations provide an indication of the wealth of patterns that the system can exhibit. In particular, periodic oscillation and spot-like patterns can be observed in one-dimensional and two-dimensional simulations, respectively. All results obtained reveal the mechanism of interaction between tumour cells and immune system, which have profound significance for the development of tumour immunotherapy.

Active manipulation of Helmholtz scalar fields: near-field synthesis with directional far-field control

In this article, we propose a strategy for the active manipulation of scalar Helmholtz fields in bounded near-field regions of an active source while maintaining desired radiation patterns in prescribed far-field directions. This control problem is considered in two environments: free space and homogeneous ocean of constant depth, respectively. In both media, we proved the existence of and characterized the surface input, modeled as Neumann data (normal velocity) or Dirichlet data (surface pressure) such that the radiated field satisfies the control constraints. We also provide a numerical strategy to construct this predicted surface input by using a method of moments-approach with a Morozov discrepancy principle-based Tikhonov regularization. Several numerical simulations are presented to demonstrate the proposed scheme in scenarios relevant to practical applications.

Solvability of Pseudoparabolic Equations with Non-Linear Boundary Condition

The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.

Observer-based finite-time output feedback control of heat equation with Neumann boundary condition

This paper is concerned with the problem of boundary observer-based finite-time output feedback control for a heat system with Neumann boundary condition and Dirichlet boundary actuator. In this system, it is endowed a single sensor at the left boundary and actuator at the right boundary. First, a backstepping finite-time observer is designed. Based on linear switching observer gains with state-dependent switching, the proposed observer estimates the system state variables in a finite-time only using one displacement boundary measurement. Then, an observer-based linear switching output feedback control with state-dependent switching is presented, which steers any solution of the output feedback closed-loop system to zero in a finite-time. Finally, the simulation results are provided to support the theoretical results.

Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions

Abstract The one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.

Grassmannian and flag sigma models on interval: phase structure and L-dependence

We discuss the two-dimensional Grassmannian SU(N)/S(U(N − 2) × U(2)) and the flag SU(N )/S(U(N − 2) × U(1) × U(1)) sigma models on a finite interval and construct analytical solutions of gap equations in the large-N limit. We show that the flag model admits a homogeneous solution for “mixed” Dirichlet-Neumann (DN) boundary conditions only for sufficiently large length L and undergoes a phase transition from the phase of partly broken gauge symmetry U(1) to the symmetric phase U(1) × U(1) for large L. On the other hand, the Grassmannian model has a detached phase with one massive and one massless non-zero condensates that completely break U(2) gauge symmetry. This phase lives on a region of L bounded from above and has to use the Robin boundary conditions. We also examine the L-dependence of the total energy and detect the linear growth inherent to confining string in all phases.

The D-N alternating algorithm based on the curved-FEM and moving mesh for the exterior harmonic problem

In this paper, we study the numerical solution of the exterior harmonic problem with the Dirichlet boundary value condition. Owing to the difficulties of solving directly in the unbounded domain, we decompose the exterior problem into a mixed boundary value problem on a bounded annular subdomain and a Dirichlet boundary value problem on an unbounded subdomain by constructing an artificial boundary. Then, the Dirichlet-Neumann (D-N) alternating algorithm is proposed to solve two sub-problems alternately, where the former is solved by the curved-FEM, while the latter is solved by the principle of natural boundary reduction (NBR). Rather than the standard FEM, the curved-FEM is conforming and gives better discrete approximate variational formulation of the algorithm. The geometrical convergence of the discrete algorithm is also obtained. Finally, the D-N alternating algorithm based on the curved-FEM and the moving mesh method is shown to achieve higher precision in the numerical example.

Speed selection for traveling waves of a reaction–diffusion–advection equation in a cylinder

This paper is concerned with linear or nonlinear selection mechanism for the minimal speed of traveling wave solutions to a reaction–diffusion–advection equation in a cylindrical domain with Fisher–KPP-type nonlinearity. By using the method of upper and/or lower solutions, we establish the speed selection results. Precisely, we obtain sufficient conditions under which the linear or nonlinear selection is realized when the model is prescribed by Neumann boundary conditions and Dirichlet boundary conditions respectively.

Inequalities between Dirichlet and Neumann eigenvalues of the polyharmonic operators

We prove that μ k + m m > λ k m \mu _{k+m}^m >\lambda _k^m , where μ k m \mu _k^m ( λ k m \lambda _k^m ) are the eigenvalues of ( − Δ ) m (-\Delta )^m on Ω ⊂ R d \Omega \subset \mathbb R^d , d ≥ 2 d\geq 2 , with Neumann (Dirichlet) boundary conditions.

Asymptotic Behavior of the Principal Eigenvalue of a Linear Second Order Elliptic Operator with Small/Large Diffusion Coefficient

In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: $-D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ { in }\Omega, $ complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition, $ \frac{\partial\phi}{\partial n}+\beta(x)\phi=0 \ \ { on }\partial\Omega, $ where $\beta\in C(\partial\Omega)$ is allowed to be positive, sign-changing, or negative, and $n(x)$ is the unit exterior normal to $\partial\Omega$ at $x$. The domain $\Omega\subset\mathbb{R}^N$ is bounded and smooth, the constants $D>0$ and $\alpha>0$ are, respectively, the diffusive and advection coefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient $D\to0$ or $D\to\infty$. Our results, together with those of [X. F. Chen and Y. Lou, Indiana Univ. Math. J., 61 (2012), pp. 45--80; A. Devinatz, R. Ellis, and A. Friedman, Indiana Univ. Math. J., 23 (1973/74), pp. 991--1011; and A. Friedman, Indiana U. Math. J., 22 (1973), pp. 1005--1015] where the Neumann boundary case (i.e., $\beta=0$ on $\partial\Omega$) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue.

About Revisiting Domain Decomposition Methods for Poroelasticity

In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different approaches, i.e., Dirichlet-Neumann (DN) and the mortar finite element method (MFEM), and we recognize their advantages and disadvantages. The MFEM significantly lessens the computational cost of reservoir compaction and subsidence calculations by dodging the conforming Cartesian grids that arise from the pay-zone onto its vicinity. There is a manifest necessity of producing non-matching interfaces between the reservoir and its neighborhood. We thus employ MFEM over nonuniform rational B-splines (NURBS) surfaces to stick these non-conforming subdomain parts. We then decouple the mortar saddle-point problem (SPP) using the Dirichlet-Neumann domain decomposition (DNDD) scheme. We confirm that this procedure is proper for calculations at the field level. We also carry comprehensive comparisons between the conventional and non-matching solutions to prove the method’s accuracy. Examples encompass linking finite element codes for slightly compressible single-phase and poroelasticity. We have used this program to a category of problems ranking from near-borehole applications to whole field subsidence estimations.

Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?

A class of Keller-Segel-Stokes systems generalizing the prototype \[ \left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \Big(n(n+1)^{-\alpha}\nabla c\Big), c_t + u\cdot\nabla c &=& \Delta c-c+n, u_t +\nabla P &=& \Delta u + n \nabla \phi + f(x,t), \qquad \nabla\cdot u =0, \end{array} \right. \qquad \qquad (\star) \] is considered in a bounded domain $\Omega\subset R^3$, where $\phi$ and $f$ are given sufficiently smooth functions such that $f$ is bounded in $\Omega\times (0,\infty)$. It is shown that under the condition that \[ \alpha>\frac{1}{3}, \] for all sufficiently regular initial data a corresponding Neumann-Neumann-Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption $\alpha>\frac{1}{2}$. In view of known results on the existence of exploding solutions when $\alpha<\frac{1}{3}$, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $\frac{1}{3}$, as previously found for the fluid-free counterpart of ($\star$), remains essentially unaffected by fluid interaction of the type considered here.

An inverse three spectra problem for Sturm\u2013Liouville operators

In this paper, we consider the inverse three spectra problems of recovering the Sturm–Liouville equation by the spectra of the Neumann–Dirichlet boundary value problem on [0,1], the Neumann–Robin problem on [0,1/2], and the Robin–Dirichlet problem on [1/2,1], where the two Robin parameters at the interior node x=1/2 are considered to be different. The algorithm of construction is presented and sufficient conditions for three sequences to be the spectral data of the mentioned boundary problems are given.

General stability of memory-type thermoelastic Timoshenko beam acting on shear force

In this paper, we consider a linear thermoelastic Timoshenko system with memory effects where the thermoelastic coupling is acting on shear force under Neumann–Dirichlet–Dirichlet boundary conditions. The same system with fully Dirichlet boundary conditions was considered by Messaoudi and Fareh (Nonlinear Anal TMA 74(18):6895–6906, 2011, Acta Math Sci 33(1):23–40, 2013), but they obtained a general stability result which depends on the speeds of wave propagation. In our case, we obtained a general stability result irrespective of the wave speeds of the system.

Boundary estimates in homogenization of systems of elasticity on Lipschitz domains: Lp theory

The primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization in Lipschitz domains. As a consequence, for d≥4, we prove that the Lp Neumann and Lp Dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2(d−1)d+1−δ<p<2+δ and 2−δ<p<2(d−1)d−3+δ respectively.

Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann–Dirichlet iterations by the Robin–Dirichlet ones which repairs the convergence for less than the first Dirichlet–Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin–Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

A one-dimensional model of flow in a junction of thin channels, including arterial trees

We study a Stokes flow in a junction of thin channels (of diameter ) for fixed flows of the fluid at the inlet cross-sections and fixed peripheral pressure at the outlet cross-sections. On the basis of the idea of the pressure drop matrix, apart from Neumann conditions (fixed flow) and Dirichlet conditions (fixed pressure) at the outer vertices, the ordinary one-dimensional Reynolds equations on the edges of the graph are equipped with transmission conditions containing a small parameter at the inner vertices, which are transformed into the classical Kirchhoff conditions as . We establish that the pre-limit transmission conditions ensure an exponentially small error , , in the calculation of the three-dimensional solution, but the Kirchhoff conditions only give polynomially small error. For the arterial tree, under the assumption that the walls of the blood vessels are rigid, for every bifurcation node a ()-pressure drop matrix appears, and its influence on the transmission conditions is taken into account by means of small variations of the lengths of the graph and by introducing effective lengths of the one-dimensional description of blood vessels whilst keeping the Kirchhoff conditions and exponentially small approximation errors. We discuss concrete forms of arterial bifurcation and available generalizations of the results, in particular, the Navier-Stokes system of equations. Bibliography: 59 titles.