Nonhomogeneous Mixed Boundary Conditions Research Articles

Existence of a solution to the non-steady magnetohydrodynamics-Boussinesq system with mixed boundary conditions

In this paper we are concerned with the non-steady magnetohydrodynamics (MHD)-Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, pressure, vorticity and stress conditions together, and in the case of static pressure may include the friction type of boundary conditions more. For the magnetic field, a non-homogeneous mixed boundary conditions are given. The conditions for temperature may include Dirichlet, Neumann and Robin conditions together. For the problem involving the static pressure and stress boundary conditions for fluid it is proved that if the data of problem are small enough, material parameters are independent of temperature, then under compatibility conditions in the initial time there exists a strong solution and the solution with small norm is unique. Much effort is paid to overcome a difficulty in dealing the advection term, Lorentz force in the momentum equation of fluid and a nonlinear term in the magnetic equation under the static pressure boundary conditions on a portion of boundary where there is flux of fluid and non-homogeneous magnetic boundary conditions. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proved without smallness of the data of problem.

Explicit structure-preserving discretization of port-Hamiltonian systems with mixed boundary control

In this contribution, port-Hamiltonian systems with non-homogeneous mixed boundary conditions are discretized in a structure-preserving fashion by means of the Partitioned FEM. This means that the power balance and the port-Hamiltonian structure of the continuous equations is preserved at the discrete level. The general construction relies on a weak imposition of the boundary conditions by means of the Hellinger-Reissner variational principle, as recently proposed in [Thoma et al., 2021]. The case of linear hyperbolic wave-like systems, including the elastodynamic problem and the Maxwell equations in 3D, is then illustrated in detail. A numerical example is worked out on the case of the wave equation.

Prandtl–Reuss dynamical elasto-perfect plasticity without safe-load conditions

In this article we study the existence theory to the Prandtl–Reuss dynamical model of elasto-perfect plasticity with non-homogeneous mixed boundary conditions. By using only the Yosida approximation of inelastic constitutive multifunction we are able to prove the existence of solutions without assuming any kind of safe-load conditions, which are quite common in the theory of elasto-perfect plasticity.

Predictor–corrector [formula omitted]- and [formula omitted]-versions of the finite element method for Poisson’s equation in polygonal domains

We consider boundary value problems for the Poisson equation on polygonal domains with general nonhomogeneous mixed boundary conditions and derive, on the one hand, explicit extraction formulas for the coefficients of the singularities. On the other hand, the formulas are used to construct efficient adaptations for the h-, p- and hp-versions of the finite element method for the numerical treatment. A priori error estimates show that the h-version of the finite element algorithm exhibits the same rate of convergence as it is known for problems with smooth solutions. However, the principal results of the present work are the robust exponential convergence results for the p- and hp-versions of the finite element method on quasiuniform meshes. In fact, it is shown that if the input data (source term and boundary data) are piecewise analytic, then with appropriate choices of conforming finite element subspaces VN of dimension N∈N, the p- and hp-versions of the finite element algorithms on quasiuniform meshes yield approximate solutions uN to the exact solution u that satisfy the estimates ‖u−uN‖H1(Ω)≤C1e−b1N23 and ‖u−uN‖H1(Ω)≤C2e−b2N13, respectively. Several numerical experiments are included to illustrate the practical effectiveness of the proposed algorithms. The results show that the theoretical error analyses are attained within the range of engineering accuracy.

Analytical Solution Based on a New Series along with the Numeric Solution of the Non-Homogeneous Transient Heat Conduction Equation, in Hollow Cylinder under General Mixed Boundary Conditions

In this study, the problem of transient conduction heat transfer for an infinite hollow cylinder under non-homogeneous mixed boundary conditions at the both surfaces simultaneously in radial direction with general heat source depend on time and radius, also general initial condition by the method of superposition and separation variables, is solved and temperature distribution is obtained analytically. A new series based on the Bessel functions is obtained for the problem of transient heat conduction without heat source by using separation of variables. Any function that has expanded conditions by the Fourier series can be expanded by this new series. Then, by expanding the heat source function according to this new series, the problem of transient heat transfer involving the thermal source has been solved and the radial temperature distribution is obtained. Due to the limited case studies, numerical solution of heat conduction equation with implicit finite difference method also is presented. Finally, a numerical example is given to compare between analytical and numerical solutions.

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L2-stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.

A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

We consider a time‐dependent and a steady linear convection‐diffusion‐reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin–Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element–finite volume method: the diffusion term is discretized by Crouzeix–Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. The ‐ and the ‐error in the unsteady case and the H1‐error in the steady one are estimated against the data, in such a way that no parameter enters exponentially into the constants involved. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1591–1621, 2016

ANALYTIC MULTIPLICATIVE SEPARATION AND EXISTENCE INVESTIGATION OF NON-LINEAR OXYGEN TRANSPORT WITH POISEUILLE HEMODYNAMIC FLOW IN A SEMI-GENERALIZED COORDINATE SYSTEM

A well known governing nonlinear PDE used to model oxygen transport is formulated in a generalized co-ordinate system where the Laplacian is expressed in metric tensor form. A reduction of the pde to simpler problem, which has never been done before, subject to specific integrability conditions is shown. A specific form of the initial equation to be reduced has been used by Nair and coworkers [1-3] describing the intraluminal problem of oxygen transport in large capillaries or arterioles and more recent work by [4] describing release of ATP in microchannels. In each of these cases, a tube with a central core, rich in red blood cells, and with a thin plasma region near the boundary wall, free of RBCs is considered. A co-ordinate transformation to solve reduced pde in core region is adopted. An extension and modification of the work of Erbe and Wang [5] is derived and existence of positive solutions of an ode obtained in the general formulation is obtained for nonhomogeneous mixed boundary conditions and on general positive interval. Finally a solution method is given to solve a linear equation used by Nair [1-3] in the plasma layer.

Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions

The numerical solution of linear elliptic partial differential equations often involves finite element discretization, where the discretized system is usually solved by some conjugate gradient method. The crucial point in the solution of the obtained discretized system is a reliable preconditioning, that is to keep the condition number of the systems under control, no matter how the mesh parameter is chosen. The PCG method is applied to solving convection–diffusion equations with nonhomogeneous mixed boundary conditions. Using the approach of equivalent and compact-equivalent operators in Hilbert space, it is shown that for a wide class of elliptic problems the superlinear convergence of the obtained preconditioned CGM is mesh independent under FEM discretization.

A nonlocal phase-field model with nonconstant specific heat

We prove the existence, uniqueness, thermodynamic consistency, global boundedness from both above and below, and continuous data dependence for a solution to an integrodifferential model for nonisothermal phase transitions under nonhomogeneous mixed boundary conditions. The specific heat is allowed to depend on the order parameter, and the convex component of the free energy may or may not be singular.

Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces

This text is devoted to maximal regularity results for second-order parabolic systems on Lipschitz domains of space dimension $n \ge 3$ with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial-value problems generates isomorphisms between two scales of Sobolev--Morrey spaces for solutions and right-hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower-order coefficients.

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.

Problems on electrorheological fluid flows

In this paper, we develop and analyze a new model describing electrorheological fluid flow. In contrast to existing models, which assume the electric field to be perpendicular to the velocity field and are thus restricted to simple shear flow and flows close to it, we consider the fluid as anisotropic and introduce a general constitutive relation based on a viscosity function that depends on the shear rate, the electric field strength, and on the angle between the electric and the velocity vectors. We study general flow problems under nonhomogeneous mixed boundary conditions with given values of velocities and surface forces on different parts of the boundary. We investigate both the case where the viscosity function is continuous and the case where it is singular for vanishing shear rate. In the latter case, the problem reduces to a variational inequality. Using methods of nonlinear analysis such as fixed point theory, monotonicity, and compactness, we establish existence results for the problems under consideration. Some efficient methods for the numerical solution of the problems are presented, and numerical results for the simulation of the fluid flow in electrorheological shock absorbers are given.

Linear Elliptic Boundary Value Problems with Non-Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non-smooth data L∞-coefficients, sets with Lipschitz boundary, regular sets, non-homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev-Campanato spaces of functions and functionals, respectively.

Series solutions for Laplace's equation with nonhomogeneous mixed boundary conditions and irregular boundaries

Series solutions to Laplace's equation provide accurate analytical results for problems with regular boundary geometries. This paper develops the series approach, providing a solution methodology for problems with nonhomogeneous, mixed boundary conditions, defined on irregular boundaries. The series coefficients are obtained by first transforming to Poisson's equation with some homogeneous boundary conditions, and then modifying the eigenfunction expansion theory to cater for irregular boundary geometry. The computational details for evaluating the series coefficients are given, and the method demonstrated by a practical example.

Optimal $H^{p,{p/2}} $ Error Estimates for a Parabolic Galerkin Method

Optimal error estimates are proved for the continuous and discrete Galerkin solution (implicit and Crank-Nicolson scheme) of a general linear parabolic equation subject to nonhomogeneous mixed boundary conditions. The norms used correspond to the nonisotropic Sobolev spaces $H^{{{(p - \alpha )}/2}} ((0,T),H^\alpha (\Omega )) \cap L^2 ((0,T),H^p (\Omega ))$ for $p \geqq \alpha \geqq - 1$. Also, estimates with respect to the norm of $\mathcal{H}^{ - 1} = $ dual space of $H^1 (\Omega )$ are mentioned.