Nonhomogeneous Dirichlet Boundary Conditions Research Articles

Boussinesq system with non-homogeneous boundary conditions

A classical stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain Ω ⊂ R 3 is considered in this paper; included is the case of a possibly disconnected boundary. We prove existence of a weak, a strong and a very weak solution in L p -theory. Uniqueness of the very weak solution is proved under a small data assumption.

Numerical analysis of a transient non-linear axisymmetric eddy current model

This paper deals with the numerical solution of an axisymmetric transient eddy current problem in a conductive non-linear magnetic media. This means that the relation between the magnetic field and the magnetic induction (i.e., the so-called B–H curve) is non-linear. We analyze a weak formulation of the resulting problem in the axisymmetric case, with the source term given by means of a non-homogeneous Dirichlet boundary condition. For its numerical approximation, we propose a fully discrete scheme based on a finite element method combined with a backward Euler time discretization. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an L2 rate of convergence of order O(h+Δt) without assuming any additional regularity of the solution. Moreover, under appropriate smoothness assumptions, we also prove an L2-like rate of convergence of order O(h2+Δt). Finally, some numerical results, which confirm the theoretically predicted behavior of the method, are reported.

ARTk−Pkapproximation for linear elasticity yielding a brokenH(div)convergent postprocessed stress

We present a non-standard mixed finite element method for the linear elasticity problem in R n with non-homogeneous Dirichlet boundary conditions. More precisely, our approach is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order k ≥ 0 for the pseudostress and piecewise polynomials of degree ≤ k for the displacement can be utilized. In addition, complementing the results in the aforementioned reference, we introduce a new postprocessing formula for the stress recovering the optimally convergent approximation of the broken H ( div ) -norm. Numerical results confirm our theoretical findings.

Space–time spectral method for the two-dimensional generalized sine-Gordon equation

In this paper, we propose a space–time spectral method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions and initial conditions. The proposed method is based on the Legendre–Galerkin spectral method in space and the spectral collocation method (or block spectral collocation method) in time. We derive a priori error estimates in both L2 and H1 norms for the semidiscrete formulation. Our formulation has spectral accuracy in both space and time. Our numerical results confirm the exponential convergence of the proposed method in both space and time.

Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions.

In this paper, we study a diffusive plant-herbivore system with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability of spatially homogeneous steady states is established. We also derive conditions ensuring the occurrence of Hopf bifurcation and steady state bifurcation. Interesting transient spatio-temporal behaviors including oscillations in one or both of space and time are observed through numerical simulations.

Strong and Mild Extrapolated $L^{2}$-Solutions to the Heat Equation with Constant Delay

We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the data, we prove a well-posedness result and give an explicit representation of solutions. Further, we prove an exponential decay rate for the energy in the dissipative case. We also show that lower order regularizations lead to ill-posedness, also for higher order equations. Finally, an application with physically relevant constants is given.

Weak solutions for an initial–boundary Q-tensor problem related to liquid crystals

The coupled Navier–Stokes and Q-tensor system is considered in a bounded three-dimensional domain under homogeneous Dirichlet boundary conditions for the velocity u and either nonhomogeneous Dirichlet or homogeneous Neumann boundary conditions for the tensor Q. The corresponding initial-value problem in the whole space R3 was analyzed in Paicu and Zarnescu (2012).In this paper, three main results concerning weak solutions will be proved: the existence of global in time weak solutions (bounded up to infinite time), a uniqueness criteria and a maximum principle for Q. Moreover, we identify how to modify the system to deduce symmetry and traceless for Q, for any weak solution. The presence of a stretching term in the Q-system plays a crucial role in all the analysis.

Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations

We exhibit a large class of Lyapunov functionals for nonlinear drift–diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero-range process and the Ginzburg–Landau dynamics, which we describe briefly. We prove, as an application, linear inequalities between such an entropy-like functional and its entropy production functional for the boundary-driven porous medium equation in a bounded domain with positive Dirichlet conditions: this implies exponential rates of relaxation related to the first Dirichlet eigenvalue of the domain. We also derive Lyapunov functions for systems of nonlinear diffusion equations, and for nonlinear Markov processes with non-reversible stationary measures.

Non-homogeneous Dirichlet boundary value problems in weighted Sobolev spaces

The paper presents existence and multiplicity results for non-linear boundary value problems on possibly non-smooth and unbounded domains under possibly non-homogeneous Dirichlet boundary conditions. We develop here an appropriate functional setting based on weighted Sobolev spaces. Our results are obtained by using global minimization and a minimax approach using a non-smooth critical point theory.

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.

3-D Finite-Element Analysis of Conductive Coupling Problems in Transmission Line Rights of Way

The 3-D finite-element method is applied to compute conductive couplings in a transmission line right of way. The domain under analysis is composed of several underground structures, such as ground wires of high-voltage towers, a coated pipeline section, and a steel-reinforced masonry enclosure. A nonhomogeneous Dirichlet boundary condition procedure is employed to overcome the modeling difficulties of this particular problem. A preliminary test problem to validate this procedure is also discussed.

Elliptic problems with nonhomogeneous boundary condition and derivatives of nonlinear terms

The paper presents existence results for nonlinear elliptic problems under a nonhomogeneous Dirichlet boundary condition. The considered elliptic equations exhibit nonlinearities containing derivatives of the solution.

Second-Order Convergence and Unconditional Stability on Crank-Nicolson Scheme for Burgers’ Equation

In this paper, we study the numerical solution of one-dimensional Burgers equation with non-homogeneous Dirichlet boundary conditions. This nonlinear problem is converted into the linear heat equation with non-homogeneous Robin boundary conditions by Hopf-Cole transformation. The heat equation is discretized by Crank-Nicolson finite difference scheme, and the fourth-order difference schemes for the Robin conditions are combined with the Crank-Nicolson scheme at two endpoints. The proposed method is proved to be second-order convergent and unconditionally stable. The numerical example supports the theoretical results.

A Petrov-Galerkin spectral method for fourth-order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth-order elliptic problems on rectangular domains subject to non-homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher-order problems. By applying these results to a fourth-order problem, we establish the H2-error and L2-error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace

We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.

LOCAL CHEBYSHEV PROJECTION–INTERPOLATION OPERATOR AND APPLICATION TO THE h–p VERSION OF THE FINITE ELEMENT METHOD IN THREE DIMENSIONS

In this paper, we construct local Chebyshev projection–interpolation operators for tetrahedral and hexahedral elements in three dimensions based on the framework of the Jacobi-weighted Sobolev and Besov spaces. A simple assembly of the local Chebyshev projection–interpolations ΠΩj u on all the elements Ωj, 1 ≤ j ≤ J, leads to a globally continuous and piecewise polynomial which possesses the best approximation properties locally and globally. By applying the local Chebyshev projection–interpolation operators to the h–p version of the finite element method with general tetrahedral or hexahedral meshes for second-order elliptic problems in three dimensions, we establish the convergence rate for problems with homogeneous Dirichlet boundary conditions and the solution [Formula: see text], and for problems with nonhomogeneous Dirichlet boundary conditions and the solution [Formula: see text].

Trajectory attractors for the Sun–Liu model for nematic liquid crystals in 3D

In this paper we prove the existence of a trajectory attractor (in the sense of Chepyzhov and Vishik) for a nonlinear PDE system obtained from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier–Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in the director equation. Finally, a dissipative estimate is obtained by using a proper integrated energy inequality. Both the cases of (homogeneous) Neumann and (non-homogeneous) Dirichlet boundary conditions for d are considered.

Analytical solution for a generalized space‐time fractional telegraph equation

In this paper, we consider a nonhomogeneous space‐time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the first‐order or second‐order time derivative by the Caputo fractional derivative , α > 0 and the Laplacian operator by the fractional Laplacian ( − Δ)β ∕ 2, β ∈ (0,2]. We discuss and derive the analytical solutions under nonhomogeneous Dirichlet and Neumann boundary conditions by using the method of separation of variables. The obtained solutions are expressed through multivariate Mittag‐Leffler type functions. Special cases of solutions are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.

Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.

Instability of an Inverse Problem for the Stationary Navier--Stokes Equations

This paper provides a theoretical study of the detection of an object immersed in a fluid when the fluid motion is governed by the stationary Navier--Stokes equations with nonhomogeneous Dirichlet boundary conditions. To solve this inverse problem, we make a boundary measurement on a part of the exterior boundary. First, we present an identifiability result. We then use a shape optimization method: in order to identify the obstacle, we minimize a nonlinear least squares. Thus, we prove the existence of the first order shape derivative of the state, characterize it, and deduce the gradient of the least squares functional. Finally, we study the stability of this setting doing a shape sensitivity analysis of order two. Hence, we prove the existence of the second order shape derivatives and we give the expression of the shape Hessian at possible solutions of the original inverse problem. Then, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the ident...