Equation In Two-dimensional Domain Research Articles

Boundary Tracing for Laplace's Equation with Conformal Mapping

The conformal mapping technique has long been used to obtain exact solutions to Laplace's equation in two-dimensional domains with awkward geometries. However, a major limitation of the technique is that it is only directly compatible with Dirichlet and zero-flux Neumann boundary conditions. It would be useful to have a means of adapting the technique to handle more general boundary conditions, for example, Robin or nonlinear flux conditions. Boundary tracing is an unconventional method for tackling boundary value problems with generic flux boundary conditions, where one takes a known solution to the field equation and seeks new boundaries satisfying the prescribed boundary condition. In this paper, we adapt boundary tracing for compatibility with conformal mapping to produce a new prescription for studying Laplace's equation coupled with general flux boundary conditions. We illustrate the procedure via two simple examples involving heat transfer. In both cases, we demonstrate how to construct infinite families of nontrivial domains in which the solution to the chosen flux boundary value problem is exactly equal to a selected harmonic function.

Existence of solutions to heat equations with hysteresis coupled with Navier–Stokes equations in 2D domains

This paper is concerned with a system of heat equations with hysteresis and Navier–Stokes equations in two-dimensional domains. There are some studies on the system of heat equations and Navier–Stokes equations in which the temperature (the solution to heat equations) is controlled by obstacle functions. In this paper, the control of the temperature is precisely described by introducing hysteresis. This paper provides the existence of solutions along the abstract theory for evolution equations with subdifferential operators.

Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains

This paper is concerned with the system of nonlinear heat equations with constraints coupled with Navier-Stokes equations in two-dimensional domains. In 2012, Sobajima, Tsuzuki and Yokota proved the existence and uniqueness of solutions to the system with heat equations including the diffusion term $\Delta\theta$, where $\theta$ represents the temperature. This paper gives the existence result in which the Laplace operator $\Delta$ is replaced with the $p$-Laplace operator $\Delta\rho$, where $p>2$.

Two-dimensional meshless solution of the non-linear convection diffusion reaction equation by the Local Hermitian Interpolation method

A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diffusion-reaction equation in two-dimensional do-mains. The Local Hermitian Interpolation (LHI) method is employed for thespatial discretization and several strategies are implemented for the solution of the resulting non-linear equation system, among them the Picard iteration, the Newton Raphson method and a truncated version of the Homotopy Analysis Method (HAM). The LHI method is a local collocation strategy in which Radial Basis Functions (RBFs) are employed to build the interpolation function. Unlike the original Kansa’s Method, the LHI is applied locally and the boundary and governing equation differential operators are used to obtain the interpolation function, giving a symmetric and non-singular collocation matrix. Analytical and Numerical Jacobian matrices are tested for the Newton-Raphson method and the derivatives of the governing equation with respect to the homotopy parameter are obtained analytically. The numerical scheme is verified by comparing the obtained results to the one-dimensional Burgers’ and two-dimensional Richards’ analytical solutions. The same resultsare obtained for all the non-linear solvers tested, but better convergence ratesare attained with the Newton Raphson method in a double iteration scheme.

Solving Integral Equations on Piecewise Smooth Boundaries Using the RCIP Method: A Tutorial

Recursively compressed inverse preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in two-dimensional domains with corners. In a series of subsequent papers, the technique was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present paper is threefold: first, to review the RCIP method in a simple setting; second, to show how easily the method can be implemented in MATLAB; third, to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.

Flux intensity functions for the Laplacian at polyhedral edges

We present explicit representation formulas for the coefficients of the singularities associated with mixed boundary value problems for the Poisson equation in two-dimensional domains with corners and three-dimensional domains with straight edges including cracks. We rely on partial Fourier analysis of the boundary value problem in the vicinity of the singularities to derive an asymptotic expansion of the solution. Hence, the edge flux intensity functions are expressed in terms of Fourier series and we give integral expressions from which the associated Fourier coefficients can be computed independently and in parallel. Our method does not require the knowledge of the dual singular solutions of the adjoint problems. Moreover, the constructive nature of the formulas provides in a straight forward way a strategy for the construction of efficient numerical methods for the accurate computation of the stress intensity factors and the edge flux intensity functions. Also, the formulas can be used to construct postprocessing algorithms for the standard finite element approximation of the solution of the boundary value problem to improve the accuracy. Examples that illustrate the efficiency and robustness of the formulas are also given.

10.1007/s12258-008-1007-7

A V-cycle multigrid method is developed for a time-dependent viscoelastic fluid flow satisfying an Oldroyd-B-type constitutive equation in two-dimensional domains. Also existence, uniqueness, and error estimates of an approximate solution are discussed. The approximate stress, velocity, and pressure are, respectively, σ k -discontinuous, u k -continuous, and p k -continuous.

A V-cycle multigrid method for a viscoelastic fluid flow satisfying an Oldroyd-B-type constitutive equation

A V-cycle multigrid method is developed for a time-dependent viscoelastic fluid flow satisfying an Oldroyd-B-type constitutive equation in two-dimensional domains. Also existence, uniqueness, and error estimates of an approximate solution are discussed. The approximate stress, velocity, and pressure are, respectively, σk-discontinuous, uk-continuous, and pk-continuous.

A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions

This paper is concerned with the mixed formulation of the Boussinesq equations in two-dimensional domains and its numerical approximation. The paper deals first with existence and uniqueness results, as well as the description of the regularity of any solution. The problem is then approximated by a mixed finite element method, where the gradient of the velocity and the gradient of the temperature, quantities of practical importance, are introduced as new unknowns. An existence result for the finite element solution and convergence results are proved near a nonsingular solution. Quasi-optimal error estimates are finally presented.

The boundary element method for the solution of Stokes equations in two-dimensional domains

A boundary element method for the solution of Stokes equations governing creeping flow or Stokes flow in the interior of an arbitrary two-dimensional domain is presented. A procedure for introducing pressure data on the boundary of the domain is also included and the integral coefficients of the resulting linear algebraic equations are evaluated analytically. Calculations are performed in a circular domain using a variety of different boundary conditions, including a combination of the fluid velocity and the pressure. Results are presented both on the boundary and inside the solution domain in order to illustrate that the boundary element method developed here provides an efficient technique, in terms of accuracy and convergence, to investigate Stokes flow numerically.

Optimal bounds on the dimension of the attractor of the Navier-Stokes equations

In this article we derive optimal upper bounds on the dimension of the attractor for the Navier-Stokes equations in two-dimensional domains, these bounds fully agree with the lower bounds obtained by Babin and Vishik (1983) (see also Ghidaglia and Temam, and Liu (1993)). As in Babin and Vishik (1983), we consider here elongated domains and leaving the density of volume forces and the viscosity fixed, we let the shape ratio of the domain become large so that the Grashof number is large. The estimates derived here are based on the general methods for estimating attractors dimensions as in Constantin et al. (1988), on a new version of the Lieb-Thirring inequalities for elongated domains and on techniques developed for such domains in RS (1993), and Temam and Ziane (1996). At the end of the article, we also give some partial results in the three-dimensional case for which we need a physical assumption on the Reynolds number introduced in Ghidaglia and Temam.

Solving the Helmholtz equation in some infinite domains

A numerical-analytical method is proposed to solve boundary-value problems for the Heimholtz equation in two-dimensional domains of complicated configuration.

A domain decomposition method for parabolic problems

In this paper, we describe a domain decomposition preconditioner used with the conjugate gradient method for solving linear systems arising from parabolic partial differential equations in two-dimensional domains. The method relies on patching local overlapping solutions and on estimates of decay of inverses of matrices to approximately compute the amount of overlapping. Numerical examples are described that show the usefulness of this method on parallel computers as its main feature is that a minimal global communication between the subdomains is required during the preconditioning phase. The number of processors that can be used in parallel is equal to the number of subdomains.

Solvability of an initial boundary value problem for the euler equations in twodimensional domain with corners

AbstractThe aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ⊂ R2 with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three‐dimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2]–[6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.