The discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines

doi:10.1016/S0045-7825(99)00046-8

Computer Methods in Applied Mechanics and Engineering

Volume 179, Issues 3–4, 2 September 1999, Pages 345-360

https://doi.org/10.1016/S0045-7825(99)00046-8 Get rights and content

Abstract

The numerical simulation for the exterior problem of Poisson equation is considered. We introduced a polygonal artificial boundary Γ_e and designed a discrete artificial boundary condition on it by using the direct method of lines. Then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary. The finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed. Furthermore numerical examples show that the discrete artificial boundary condition is very effective and more accurate than the Neumann boundary condition which is often used in engineering literatures.

Introduction

When computing the numerical solution of a boundary value problem of partial differential equations in an unbounded domain, one often introduces an artificial boundary to cut off the unbounded part of the domain and sets up an artificial boundary condition at the artificial boundary. Then the original problem is reduced to a boundary value problem defined in a bounded computational domain. In order to limit the computational cost, the bounded computational domain must not be too large. Then the artificial boundary condition must be a good approximation of the exact boundary condition on the artificial boundary. Thus the accuracy of the artificial boundary condition and the computational cost are closely related. Therefore designing an artificial boundary condition with high accuracy on a given artificial boundary has become an effective and important method for solving partial differential equations in an unbounded domain which arises in various fields of engineering, such as fluid flow around obstacles, coupling of structures with foundation, wave propagation and so on.

There are many authors who have worked on this subject for various problems by different techniques. For instance, Engquist and Majda [1] designed absorbing boundary conditions for the wave equation. Goldstein [2] presented the exact boundary condition and a sequence of its approximations at an artificial boundary for Helmholtz-type equation in waveguides. Feng [3] proposed the asymptotic radiation conditions for the reduced wave equation by using the asymptotic approximation of Hankel functions. Han and Wu [4], [5] obtained the exact boundary conditions and a series of their approximations at an artificial boundary for the Laplace equation and the linear elastic system. The exact boundary condition at an artificial boundary for partial differential equations in an infinite cylinder was proposed by Hagstrom and Keller [6], [7]. Shortly after, they used this technique to solve nonlinear problems. A family of artificial boundary conditions for unsteady Oseen equations in the velocity pressure formulation with small viscosity was developed by Halpern and Schatzman [8], which was then applied to unsteady Navier–Stokes equations. Nataf [9] designed an open boundary condition for a steady Oseen equation in streamfunction vorticity formulation, which is applied to viscous incompressible fluid flow around a body in a flat channel with slip boundary conditions on the wall. Hagstrom [10], [11] proposed asymptotic boundary conditions at an artificial boundary for the simulation of time-dependent fluid flows. Han et al. [12] designed discrete artificial boundary conditions for incompressible viscous flows in an infinite channel by using a fast iterative method. Han and Bao [13], [14] proposed discrete artificial boundary conditions for incompressible viscous flows in a channel by using the method of lines. Han et al. [15] developed artificial boundary conditions for the problem of infinite elastic foundation. Recently Ben-Porat and Givoli [16] considered an elliptic artificial boundary for the Laplace equation. One can find more references in Ref. [17].

Because of the restriction of the techniques they used, many authors mainly consider the regular artificial boundaries, such as circumferences, straight lines or a segment, as artificial boundaries in solving two-dimensional problems. As we know, it is very easy to implement discretizing a boundary value problem in a bounded domain with a polygonal boundary by using the finite element method [18]. Thus from the engineering point of view, it is natural to introduce a polygonal boundary as an artificial boundary for the problem in an unbounded domain. Thus how to design an artificial boundary condition with high accuracy at a given polygonal artificial boundary becomes an interesting open problem. In this paper, we propose a method to design a discrete artificial boundary condition at a given polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines. Then the problem is reduced to a boundary value problem defined in a bounded computational domain. Finite element approximation of the reduced problem is also considered. Furthermore numerical examples show that the discrete artificial boundary condition presented in this paper is very effective.

Section snippets

The discrete artificial boundary condition at a polygonal artificial boundary

Let Γ_i be a bounded, simple and closed curve in $R^{2}$ and $Ω$ be the unbounded domain with boundary Γ_i. We consider the following model problem, the exterior problem of Poisson equation: $Δu=f in Ω,$ $u|_{Γ_{i}} =g,$ $u is bounded when r→+∞;$ where g is a given function on Γ_i, f is a given function in $Ω$ and we assume that its support is compact. This problem is defined in the unbounded domain $Ω$ . In Ref. [4], this problem was considered. They introduced a circumference as an artificial boundary and designed a series of

The numerical solution of the problem (2.1)–(2.3)

On the bounded computational domain $Ω_{i}$ , we now consider the numerical solution of the problem (2.1)–(2.3). As we know that the restriction of u, the solution of the problem (2.1)–(2.3), satisfies the boundary value problem (2.9)–(2.11). Let $H^{1} (Ω_{i})$ denote the usual Sobolov space on $Ω_{i}$ [19] and assume that $T_{g} ={v∈H^{1} (Ω_{i}) ∣v∣_{Γ_{i}} =g}, T_{0} ={v∈H^{1} (Ω_{i}) ∣v∣_{Γ_{i}} =0}.$ Then the boundary value problem (2.9)–(2.11) is equivalent to the following variational problem: $Find u∈T_{g} such that$ $a(u,v)+b(u,v)=f(v) ∀v∈T_{0},$ with $a(u,v)=∫_{Ω_{i}} ∇u$

Numerical implementation and examples

Let $Ω$ denote the exterior domain of the unit square, namely $Ω={x=(x_{1},x_{2}) | |x_{1} |>1 or |x_{2} |>1}.$ We consider the numerical solution of the original problem (2.1)–(2.3) with given f and g. We take the artificial boundary $Γ_{e} ={(x_{1},x_{2}) | x_{1} =±2, −2⩽x_{2} ⩽2}∪{(x_{1},x_{2}) | x_{2} =±2, −2⩽x_{1} ⩽2}$ . Hence $Ω_{e} ={(x_{1},x_{2}) | |x_{1} |>2 or |x_{2} |>2}$ and $Ω_{i} =Ω⧹ Ω ̄_{e}$ . Since the solution of each example, u(x₁,x₂), is symmetric about x₂ axes and antisymmetric about x₁ axes, respectively, the domain of computation is taken to be the part of $Ω_{i}$ lying in the

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^☆: This work was supported partly by the Climbing Program of National Key Project of Foundation, Doctoral Program foundation of Institution of Higher Education and the National Natural Science Foundation of China. Computation was supported by the State Key Lab. of Scientific and Engineering Computing in China.

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The discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines☆

Abstract

Introduction

Section snippets

The discrete artificial boundary condition at a polygonal artificial boundary

The numerical solution of the problem (2.1)–(2.3)

Numerical implementation and examples

J. Comput. Phys.

J. Comput. Phys.

Comput. Methods Appl. Mech. Engrg.

Absorbing boundary conditions for the numerical simulation of waves

Math. Comp.

A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains

Math. Comp.

Asymptotic radiation conditions for reduced wave equations

J. Comput. Math.

Approximation of infinite boundary condition and its application to finite element method

J. Comput. Math.

The approximation of exact boundary condition at an artificial boundary for linear elastic equation and its application

Math. Comp.

Exact boundary conditions at artificial boundary for partial differential equations in cylinders

SIAM J. Math. Anal.

Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains

Math. Comp.