Multigrid convergence for a singular perturbation problem

So far there has been no analysis of multigrid methods applied to singularly perturbed Dirichlet boundary-value problems. Only for periodic boundary conditions does the Fourier transformation (mode analysis) apply, and it is not obvious that the convergence results carry over to the Dirichlet case, since the eigenfunctions are quite different in the two cases. In this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the Dirichlet case.     

An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

     

Stabilized boundary element methods for exterior Helmholtz problems

In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries. The proposed regularization is done based on boundary integral operators which are already included in standard boundary element formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations.     

Nonlinear boundary value problems with concave nonlinearities near the origin

In this paper we consider the effect of concave nonlinearities for the solution structure of nonlinear boundary value problems such as Dirichlet and Neumann boundary value problems of elliptic equations and periodic boundary value problems for Hamiltonian systems and nonlinear wave equations.     

Periodic and Almost Periodic Solutions for a Coupled System of Two Korteweg-de Vries Equations with Boundary Forces

In this paper, we investigate a coupled system of two Korteweg-de Vries equations on a bounded domain. We discuss the long-time behavior of this system with forces on the left Dirichlet boundary conditions. We obtain that if the forces are periodic (almost periodic) with small amplitude, then the solution of the coupled system is periodic (almost periodic).     

On homogenization of solutions of the poisson equation in a perforated domain with different types of boundary conditions on different cavities

In this paper we consider boundary value problems in perforated domains with periodic structures and cavities of different scales, with the Neumann condition on some of them and mixed boundary conditions on others. We take a case when cavities with mixed boundary conditions have so called critical size (see [1]) and cavities with the Neumann conditions have the scale of the cell. In the same way other cases can be studied, when we have the Neumann and the Dirichlet boundary conditions or the Dirichlet condition and the mixed boundary condition on the boundary of cavities.There is a large literature where homogenization problems in perforated domains were studied [2];-[7];     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.     

Iterative Methods for Boundary Integral Equations

We review some iterative methods for solving boundary integral equations which arise in Dirichlet and Neumann problems for the Helmholtz and Laplace equations. In particular we show how these integral equations may be transformed so that they may be solved by Neumann-Poincare Picard iteration.     

Existence and stability of periodic solutions for competing species diffusion systems with dirichlet boundary conditions

The article considers positive t-periodic solutions for a periodic system of competing-species diffusion-reaction equations with zero or positive Dirichlet boundary conditions. The asymptotic orbital stability of the periodic solution is also investigated. Some results are applicable to cases when interspecies interactions are not small     

Global existence of solutions for non‐small data to non‐linear spherically symmetric thermoviscoelasticity

We consider some initial–boundary value problems for non‐linear equations of thermoviscoelasticity in the three‐dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd.     

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.The research of this author was supported by an SERC Grant.     

H?lder estimates for elliptic equations degenerate on part of the boundary of a domain

The present paper studies the Dirichlet problem for elliptic equations degenerate on part of the boundary of a domain and the degeneracy is of the Keldysh type. By introducing a proper metric that is related to the operator we establish the global H?lder estimates when some well-posed boundary conditions are satisfied. The main methods are the construction of some barrier functions and the interpolation of the estimates of uniformly elliptic operators.     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

On the Solution of the Exterior Dirichlet Problem for an Axisymmetric Potential

For an unbounded domain of the meridian plane with bounded smooth boundary that satisfies certain additional conditions, we develop a method for the reduction of the Dirichlet problem for an axisymmetric potential to Fredholm integral equations. In the case where the boundary of the domain is a unit circle, we obtain a solution of the exterior Dirichlet problem in explicit form.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Variational approach to some damped Dirichlet problems with impulses

In this paper, we study the existence of solutions for damped nonlinear impulsive differential equations with Dirichlet boundary conditions. By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution. Some recent results are extended and significantly improved. Finally, some examples are presented to illustrate our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Darcy equation for random porous media

Many attempts have been made recently to deduce Darcy equations for flows through porous media. Much has been done for models having a periodic microstructure. This paper presents a mathematical analysis based on the more general and more realistic assumption that the microstructure of porous media is random and stochastically homogeneous. For this type of random domain we consider homogenization of the Poisson and Stokes equations supplemented by homogeneous Dirichlet boundary conditions on the random boundary. With this approach we get the Darcy equation in general as well as present details for the particular case of a checkerboard model of porous media. © 1996 John Wiley & Sons, Inc.     

Boundary element method for spatially-periodic potential problems

In this paper, we propose a boundary element method for two-dimensional potential problems with one-dimensional spatial periodicity, which have been difficult to be solved by the ordinary boundary element method. In the presented method, we reduce the potential problems with Dirichlet and Neumann boundary conditions to integral equation problems with the periodic fundamental solution of the Laplace operator and, then, obtain approximate solutions by solving linear systems given by discretizing the integral equations. Numerical examples are also included. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On existence of periodic solutions for gradient systems with applications

In this paper we prove the existence of periodic solutions for gradient systems in finite and infinite dimensional spaces. The techniques of the proofs are based on the application of a global inverse functions theorem, the Schäefer fixed point theorem and the Faedou–Galerkin method. We apply our results in order to solve nonlinear reaction–diffusion equations with Dirichlet and Neumann boundary conditions.