A parameter robust numerical method for a nonlinear system of singularly perturbed elliptic equations

https://doi.org/10.1016/j.cam.2020.113017Get rights and content

Abstract

This paper deals with a parameter robust numerical method for solving a nonlinear singularly perturbed reaction–diffusion system of elliptic equations. Existence and uniqueness of a solution to the nonlinear system of elliptic equations are presented. Existence and uniqueness of a solution to a nonlinear difference scheme, which approximates the nonlinear elliptic system, are established. The uniform convergence of the nonlinear difference scheme on piecewise uniform and log-meshes to the solution of the nonlinear elliptic system is established. A monotone iterative method for solving the nonlinear difference scheme is given. The uniform convergence of monotone iterates to the solutions of the nonlinear difference scheme and to the nonlinear elliptic system is proved. Numerical experiments confirm the established theoretical results.

Introduction

We are interested in solving the nonlinear reaction–diffusion system of elliptic equations μ2Δuk+fk(x,y,u)=0,(x,y)ω=0<x<1,0<y<1,uk(x,y)=gk(x,y),(x,y)ω,k=1,,K, where u=(u1,,uK), μ is a small positive parameter and ω is the boundary of ω. The functions fk and gk, k=1,,K, are sufficiently smooth in their respective domains. For μ1, problem (1) is singularly perturbed and characterized by the boundary layers of width O(μ|lnμ|) at the boundary ω (see Section 2.3 for details).

It is well-known that classical numerical methods for solving singularly perturbed problems are inefficient, since in order to resolve layers they require a fine mesh covering the whole domain. The basic property of the efficient numerical methods is uniform convergence with respect to the perturbation parameter, that is, numerical methods which generate μ-uniformly convergent numerical approximations to the solution. The three books [1], [2] and [3] develop these approaches and give comprehensive applications to wide classes of singularly perturbed problems.

Numerical methods for singularly perturbed linear reaction–diffusion problems have received much attention in recent papers. A brief summary of previous work on the numerical solution of singularly perturbed single and coupled linear reaction–diffusion problems can be found in [4], [5]. To the best of our knowledge, no published paper considers a system of singularly perturbed nonlinear reaction–diffusion problems posed on a two-dimensional polygonal domain.

Our goal is to construct a μ-uniform numerical method for solving problem (1). In [6], for solving a singularly perturbed single nonlinear elliptic reaction–diffusion problem, we investigate uniform convergence properties of a nonlinear finite difference method based on layer-adapted meshes. In this paper, we extend our investigation from [6] to the case of the system of singularly perturbed nonlinear elliptic reaction–diffusion problems posed on the two-dimensional polygonal domain (1).

The structure of the paper is as follows. In Section 2, properties of solutions of the nonlinear elliptic system (1) are discussed. In Section 3, we construct the nonlinear difference scheme which approximates (1). Section 3 deals with the uniform convergence of the nonlinear difference scheme on the piecewise uniform and log-meshes. In Section 5, we prove the uniform convergence of a monotone iterative method which solves the nonlinear difference scheme. The numerical experiments are presented in Section 6. The accuracy is investigated for two types of layer-adapted meshes. The numerical data confirm our theoretical results on the accuracy of the numerical method and convergence order estimates. Section 7 deals with some concluding remarks.

Section snippets

Assumptions on the nonlinear reaction functions

Some hypotheses must be placed on the nonlinear reaction functions fk, k=1,,K, in order to ensure that the nonlinear system (1) has solutions. By using the notation σ¯ω¯×(,), we suppose that fk, k=1,,K, satisfy the following assumptions: 0<c̲k(x,y)fkuk(x,y,u)c¯k(x,y),(x,y,u)σ¯,k=1,,K,mink=1,,Kmin(x,y,u)σ¯fkuk(x,y,u)=α>0,0fkuk(x,y,u)qkk(x,y),(x,y,u)σ¯,kk,k,k=1,,K,0<β=maxk=1,,Kmax(x,y)ω¯kkqkk(x,y)c̲k(x,y)<1, where c̲k, c¯k, and qkk, k,k=1,,K, kk are

The nonlinear difference scheme

On ω¯ introduce a rectangular mesh ω¯h=ω¯hx×ω¯hy: ω¯hx=xi,0iNx;x0=0,xNx=1;hxi=xi+1xi,ω¯hy=yj,0jNy;y0=0,yNy=1;hyj=yj+1yj. To discretize problem (1), we use the standard central difference operator LhUk(p)+fk(p,U)=0,pωh,Uk(p)=gk(p),pωh,k=1,,K, where U=(U1,,UK), p=(xi,yj). The linear operator Lh is defined by LhV=μ2Dx2V+Dy2V,where Dx2V and Dy2V are the central difference approximations to the second derivatives of a scaler mesh function V(p) Dx2Vij=ħxi1Vi+1,jVijhxi1VijVi1,jhx,i1

Truncation errors of the solution to the nonlinear system

Fix xi,yjωh, and introduce the one-dimensional differential equation in the space variable x, μ2d2ukx,yjdx2=ψk(x)x,yj,xi<x<xi+1,ψk(x)x,yj12fkx,yj,ux,yj,k=1,,K. Using Green’s function Gi(x) of the differential operator μ2d2dx2 on [xi,xi+1], we represent the exact solution uk(x,yj), k=1,,K, in the form ukx,yj=ukxi,yjϕ1i(x)+ukxi+1,yjϕ2i(x)+xixi+1Gi(x)(x,s)ψk(x)s,yjds,k=1,,K, where the local Green function Gi(x) is given by Gi(x)(x,s)=1μ2wi(x)(s)ϕ1i(s)ϕ2i(x),xs;ϕ1i(x)ϕ2i(s),xs,wi(x)(s)=ϕ2i(

Uniform convergence of monotone iterates

For solving the nonlinear difference scheme (9), we consider the monotone iterative method from [10].

Two vector mesh functions U˜(p)=(U˜1(p),,U˜K(p)),Û(p)=(Û1(p),,ÛK(p)),pω¯h,are called upper and lower solutions to (9), if they satisfy the inequalities LhÛk(p)+fk(p,Û)0LhU˜k(p)+fk(p,U˜),pωh,Ûk(p)g(p)U˜k(p),pωh,k=1,,K. The monotone iterative method from [10] is given in the following form: (Lh+c¯k(p))Zk(n)(p)=Rk(p,U(n1)),pωh,Rk(p,U(n1))=LhUk(n1)(p)+fk(p,U(n1)),Z(n)(p)=U(n)(p)

Numerical experiments

Here, we present numerical experiments with the special fitted meshes (17), (20), where the nonlinear difference problem (9) is solved by the monotone iterative method (23) from [10]. For our test problems, exact solutions are unknown, and numerical solutions are compared to corresponding reference solutions. We investigate the numerical error and numerical order of convergence with respect to N1, Nx=Ny=N. We define the numerical error EN as follows EN=maxk=1,,KUkNUkrefω¯h,where Uref is

Conclusion

In this paper, we have extended our investigation from [6] to the nonlinear singularly perturbed elliptic system of the reaction–diffusion type.

We have established the existence and uniqueness of the solution of the nonlinear difference scheme (9) which approximates the nonlinear elliptic problem (1). We have proved that the nonlinear difference scheme on the piecewise uniform mesh of the Shishkin type and on the log-mesh converges μ-uniformly to the solution of the nonlinear singularly

References (13)

  • BoglaevI.

    A parameter uniform numerical method for a nonlinear elliptic reaction–diffusion problem

    J. Comput. Appl. Math.

    (2019)
  • FarrellP.A. et al.

    Robust Computational Techniques for Boundary Layers

    (2000)
  • MillerJ.J.H. et al.

    Fitted Numerical Methods for Singular Perturbation Problems

    (1996)
  • RoosH.-G. et al.

    Numerical Methods for Singularly Perturbed Differential Equations

    (2008)
  • KellogR.B. et al.

    A parameter-robust numerical method for a system of reaction–diffusion equations in two dimensions

    Numer. Methods Partial Differential Equations

    (2008)
  • LinssT. et al.

    Numerical solution of systems of singularly perturbed differential equations

    Comput. Methods Appl. Mech. Eng.

    (2009)
There are more references available in the full text version of this article.

Cited by (0)

View full text