# Exact Artificial Boundary Conditions for Quasi-Linear Problems in Semi-Infinite Strips

Authors: Yajun Chen; Qikui Du
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, the exact artificial boundary conditions for quasi-linear problems in semi-infinite strips are investigated. Based on the Kirchhoff transformation, the exact and approximate boundary conditions on a segment artificial boundary are derived. The error estimate for the finite element approximation with the artificial boundary condition is obtained. Some numerical examples show the efficiency of this method.


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## Body

## 1. Introduction

The quasi-linear problems in semi-infinite strips have many physical applications in the field of magnetostatics or compressible flow around an obstacle in a channel. There have been many relevant works about quasi-linear problems in bounded domains, for example, the Galerkin approximations [1, 2], the finite element method [3, 4], and the mixed finite element method [5–7] for quasi-linear problems. One can refer to [8–10], for more related works.The artificial boundary method [11, 12], which is also called coupling of the finite element method with natural boundary reduction [13–15] or the DtN method [16, 17], is a common method to deal with quasi-linear problems in unbounded domains. In the last decade, artificial boundaries of various shapes have been derived for quasi-linear problems in unbounded domains. Circular [18, 19] and elliptical [20] artificial boundaries are for two-dimensional problems, spheroidal artificial boundaries [18] are for three-dimensional problems, and circular arc artificial boundaries [21] are for problems in concave angle domains.The purpose of this paper is to propose an artificial boundary method of using a segment artificial boundary for quasi-linear problems in semi-infinite strips. The segment artificial boundary we proposed in this paper is different with the circular artificial boundary in [18]. We also obtain an error estimate in Section 3, which was not discussed in [18].LetΩ be a strip, and b is the width of the channel Ω. The boundary of domain Ω is decomposed into three disjoint parts: ΓW,ΓN, and ΓS (see Figure 1). We introduce a Cartesian coordinate system x1,x2, such that the ray ΓS coincides with the x1-axis.Figure 1
The illustration of domainΩ.We consider the following quasi-linear problem:(1)−∂∂x1ax,u∂u∂x1+∂∂x2ax,u∂u∂x2=f,inΩ,∂u∂n=0,onΓNUΓS,u=0,onΓW,uxis bounded,asx1⟶+∞,where n is the unit exterior normal vector on ΓN or ΓS and ax,u and f are two given functions.Suppose that∂a/∂s and ∂2a/∂s2 are continuous, and a⋅,⋅ satisfies [1](2)0<C0≤ax,u≤C1,∀u∈ℝ, and for almost allx∈Ω,where C0andC1 are two positive constants, and(3)ax,u−ax,v≤CLu−v,∀u,v∈ℝ, and for almost allx∈Ω,where CL>0 is a positive constant. We also assume that f∈L2Ω has compact support, i.e., there exists a constant d0>0, such that(4)suppf⊂Ωd0=x∈Ω|x1≤d0.Additionally, we suppose that(5)ax,u≡a˜u,whenx1≥d0.The rest of the paper is organized as follows. In Section2, we derive the exact artificial boundary condition on a segment. In Section 3, we discuss the finite element approximation and a new error estimate. In Section 4, we give some numerical examples to show the efficiency of the method. The conclusions are given in Section 5.


## 2. Exact Quasi-Linear Artificial Boundary Condition

We introduce a segment artificial boundaryΓE=x1,x2|x1=d,0≤x2≤b to enclose suppf, which divides Ω into a bounded domain ΩW and an unbounded domain ΩE (see Figure 2).Figure 2
The illustration of domainΩW and ΩE.Then, the original problem (1) can be described in the coupled form:(6)−∂∂x1ax,u∂u∂x1+∂∂x2ax,u∂u∂x2=f,inΩW,∂u∂n=0,onΓNWUΓSW,u=0,onΓW,(7)−∂∂x1a˜u∂u∂x1+∂∂x2a˜u∂u∂x2=0,inΩE,∂u∂n=0,onΓNEUΓSE,uxis bounded,asx1⟶+∞,(8)uxanda˜u∂u∂nare continuous on the artificial boundaryΓE,where ΓNW=ΓN∩ΩW, ΓSW=ΓS∩ΩW, ΓNE=ΓN∩ΩE, and ΓSE=ΓS∩ΩE.We introduce the Kirchhoff transformation [22]:(9)wx=∫0uxa˜ξdξ,forx∈ΩE.Sincea˜u is a positive function, transformation (9) is invertible. Notice that(10)∇w=a˜u∇u.Then, quasi-linear problem (1) can be transformed into a linear problem as follows:(11)−Δw=0,inΩE,∂w∂n=0,onΓNEUΓSE,wxis bounded,asx1⟶+∞.By the natural boundary reduction [13–15], we know that the solution of problem (11) has the Fourier series expansion:(12)wx1,x2=a02+∑n=1+∞aned−x1nπ/bcosnπx2b,where(13)an=2b∫0bwd,x2′cosnπx2′bdx2′,n=0,1,2,⋯.We differentiate (12) with respect to x1 and set x1=d to obtain(14)∂w∂x1x1,x2|x1=d=−2πb2∑n=1+∞n∫0bwd,x2′cosnπx2′bcosnπx2bdx2′Since(15)a˜u∂u∂n=∂w∂n=−∂w∂x1,we have the exact artificial boundary condition of u on ΓE:(16)a˜u∂u∂n=2πb2∑n=1+∞n∫0b∫0ud,x2′a˜ξdξcosnπx2′bcosnπx2bdx2′≜Kud,x2.By the exact artificial boundary condition (16), we obtain(17)−∂∂x1ax,u∂u∂x1+∂∂x2ax,u∂u∂x2=f,∈inΩW,∂u∂n=0,onΓNWUΓSW,u=0,onΓW,a˜u∂u∂n=Kud,x2,onΓE.SupposeV=v∈H1ΩWvΓW=0; then, problem (17) is equivalent to the following variational problem:(18)Findu∈V,such that,Au;u,v+Bu;u,v=Fv,∀v∈V,where(19)Aw;u,v=∫ΩWax,w∂u∂x1∂v∂x1+∂u∂x2∂v∂x2dx,(20)Bw;u,v=∑n=1+∞2nπ∫0b∫0ba˜wd,x2′∂u∂x2′d,x2′∂v∂x2d,x2sinnπx2′bsinnπx2bdx2′dx2,(21)Fv=∫ΩWfxvxdx.For anys∈ℝ, we have the following equivalent definition of Sobolev spaces HsΓE [23]:(22)∀HsΓE⇔vd,x2=c02+∑n=1+∞cncosnπx2bandc022+∑n=1+∞1+n2scn2<∞.The norm ofHsΓE can be defined as follows:(23)vd,x2s,ΓE=c022+∑n=1+∞1+n2scn21/2.Then, we obtain the following lemma.Lemma 1.
The bilinear formBu;u,v is symmetric, continuous, and semidefinite on V×V.Proof.
Foru,v∈V, we suppose that(24)ud,x2′=a02+∑n=1+∞ancosnπx2′b,vd,x2=c02+∑n=1+∞cncosnπx2′b.
Taking the derivative with respect tox2′ and x2, we obtain(25)∂u∂x2′d,x2′=∑n=1+∞nπbansinnπx2′b,∂v∂x2d,x2=∑n=1+∞nπbcnsinnπx2b.
Then, we have(26)Bu;u,v≤C∑n=1+∞nan21/2∑n=1+∞ncn21/2≤Cu1/2,ΓEv1/2,ΓE≤Cu1,ΩWv1,ΩW.
Next, we show thatBu;u,v is semidefinite. For any given v∈V, we consider the auxiliary problem as follows:(27)−∂∂x1ax,u∂u∂x1+∂∂x2ax,u∂u∂x2=0,inΩE,∂u∂n=0,onΓNEUΓSE,u=v,onΓE,uxis bounded,asx1⟶+∞.
The solutionu of the above problem satisfies(28)a˜u∂u∂n=Kud,x2.
We multiply (27) by u and integrate over ΩE; then, we have(29)Bu;u,u=∫ΩEa˜u∇u2dx≥0.
This completes the proof.
In practice, we have to truncate the infinite series in (16) by finite terms; let(30)KNu=2πb2∑n=1Nn∫0b∫0ud,x2′a˜ξdξcosnπx2′bcosnπx2bdx2′.
Consider the approximation problem(31)−∂∂x1ax,uN∂uN∂x1+∂∂x2ax,uN∂uN∂x2=f,inΩW,∂uN∂n=0,onΓNWUΓSW,uN=0,onΓW,a˜uN∂uN∂n=KNuN,onΓE.
Problem (31) is equivalent to the following variational problem:(32)FinduN∈V,such that,AuN;uN,v+BNuN;uN,v=Fv,∀v∈V,where(33)BNw;u,v=∑n=1N2nπ∫0b∫0ba˜wd,x2′∂u∂x2′d,x2′∂v∂x2d,x2sinnπx2′bsinnπx2bdx2′dx2.
Similar with Lemma1, we haveLemma 2.
The bilinear formBNu;u,v is symmetric, continuous, and semidefinite on V×V.


## 3. Finite Element Approximation

Suppose thatJh is a quasi-uniform and regular triangulation of ΩW such that(34)ΩW=UK∈JhK,where K is a (curved) triangle and h is the maximal diameter of the triangles. Let(35)Vh=vh∈V,vh|Kis a linear polynomial,∀K∈Jh.We consider the approximation problem of (32):(36)FinduhN∈Vh,such thatAuhN;uhN,vh+BNuhN;uhN,vh=Fvh,∀vh∈Vh.Theorem 1.
The variational problems (18), (32), and (36) are uniquely solvable.Proof.
By (2), we have(37)Au;v,v≥C0v1,ΩW2,Au;u,v≤C1u1,ΩWv1,ΩW.
This means thatAu;u,v is coercive and bounded in V. From Lemma 1, we obtain that Au;u,v+Bu;u,v is also coercive and bounded in V. By (3), we get that ax,u is uniformly Lipschitz continuous with respect to u. Under these conditions, referring to [1], we obtain that variational problem (18) has a unique solution u∈V, for all f∈L2Ω. It is easy to deduce that problems (32) and (36) are uniquely solvable in the same way.
We assumeu,uN∈H2ΩW and uhN∈Vh are the solutions of problems (18), (32), and (36), respectively. We also suppose that(38)Vh⊂V∩W1,2+εΩWfor someε∈0,1.
Additionally, we require thatVhh⟶0 is a family of finite-dimensional subspaces of V∩CΩW, such that, for any(39)v∈V∩CΩW,there existsvh:vh∈Vh,v−vh⟶0,ash⟶0,(40)vh1,2+ε,ΩW≤Cv,for anyh,where Cv>0 is independent of h.
Then, we obtain that the continuous piecewise polynomial spaces (35) satisfy condition (38). Moreover, if we assume vh=Πhv, where Πh:v⟶vh is the interpolation operator, then, by (40), we obtain(41)vh1,2+ε,ΩW≤Πhv−v1,2+ε,ΩW+v1,2+ε,ΩW≤Cv.
Following the convergence theory in [4, 15], we have the result as follows:(42)limh⟶0uhN−uN1,ΩW=0anduN⊂V∩W1,2+εΩW,∀N≥0.
Furthermore, we have the following lemma.Lemma 3.
Supposeu is the solution of (18) and uN is the solution of (32); we have(43)limN⟶+∞u−uN1,ΩW=0.Proof.
From (2) and Theorem 2, we have(44)uN1,ΩW2≤CAuN;uN,uN+BuN;uN,uN=CFuN+BuN;uN,uN−BNuN;uN,uN≤Cf0,ΩWuN1,ΩW+BuN;uN,uN−BNuN;uN,uN.
ForuN∈V, we suppose(45)wNx1,x2′=∫0uNx1,x2′a˜ξdξ=a02+∑n=1+∞aned0−x1nπ/bcosnπx2′b,∀x1>d0,uNd,x2=c02+∑n=1+∞cncosnπx2b.
Then,(46)BuN;uN,uN−BNuN;uN,uN=∑n=N+1+∞2nπ∫0b∫0b∂wN∂x2′d,x2′∂uN∂x2d,x2sinnπx2′bsinnπx2bdx2′dx2=∑n=N+1+∞nπ2ed0−dnπ/bancn≤Ced0−dN+1π/b∑n=N+1+∞nan21/2∑n=N+1∞ncn21/2≤Ced0−dN+1π/bwN1/2,Γd0uN1/2,ΓE≤Ced0−dN+1π/buN1,ΩW2.
Fromd>d0, we have that uN is bounded in V. So, there exists a subsequence uNn, s.t. uNn⇀u¯∈V. Then, similar with Lemma 3,4 in [18], we obtain (43).
Then, we have the following convergence theorem.Theorem 2.
Letu∈H2ΩW, and assumptions (38)–(40) be satisfied. Then,(47)limh⟶0,N⟶+∞u−uhN1,ΩW=0.Next, we deduce the error estimates. We suppose that the solutionu of problem (1) satisfies(48)u|ΩW∈V∩Wk,2+εΩW,ε>0,k≥2.For simplicity, we also define some notations as follows:(49)Du;u,v≜Au;u,v+Bu;u,v,DNuN;uN,v≜AuN;uN,v+BNuN;uN,v,DNuhN;uhN,vh≜AuhN;uhN,vh+BNuhN;uhN,vh.Then, problems (18), (32), and (36) can be replaced by some simple forms, respectively. Moreover, we introduce the following bilinear form D′u;⋅,⋅ and DN′uN;⋅,⋅(50)D′u;v,z=∫ΩW∂a∂sx,uv∇u⋅∇zdx+∫ΩWax,u∇v⋅∇zdx+∫0b∫0b∂a˜∂suv∂u∂x2′d,x2′∂z∂x2d,x2∑n=1+∞2nπsinnπx2′bsinnπx2bdx2′dx2+∫0b∫0ba˜u∂v∂x2′d,x2′∂z∂x2d,x2∑n=1+∞2nπsinnπx2′bsinnπx2bdx2′dx2,DN′uN;v,z=∫ΩW∂a∂sx,uNv∇uN⋅∇zdx+∫ΩWax,uN∇v⋅∇zdx+∫0b∫0b∂a˜∂suNv∂uN∂x2′d,x2′∂z∂x2d,x2∑n=1N2nπsinnπx2′bsinnπx2bdx2′dx2+∫0b∫0ba˜uN∂v∂x2′d,x2′∂z∂x2d,x2∑n=1N2nπsinnπx2′bsinnπx2bdx2′dx2.SupposeV' is the dual space of V. By (2) and continuity of ∂a/∂s⋅,u⋅, we obtain that D′u;⋅,⋅ is bounded on V×V. Then, there exists an operator T:V⟶V′ such that(51)Tv,z=D′u;v,z,∀v,z∈V.Similar to Lemma 2.2 in [19], we have the lemma as follows:Lemma 4.
The following inequality,(52)Tv,v+Kv0,ΩW2+v1/2,ΓE2≥Cv1,ΩW2,∀v∈V,holds, where K≥0 is a sufficient large constant.
We suppose that(53)D′u;v,z=0,∀z∈V⇒v=0.AssumeI:V⟶V′ is the canonical injection. Since V is compactly embedded in L2ΩW, we obtain that the operator J:V⟶V′ defined by Jv=Iv,0 is also compact. Then, we deduce that T:V⟶V′ is an isomorphism.By conditions (19), (52), and (53) and Theorem 10.1.2 in [24], there existsh0∈0,1, s.t. the following inf-sup condition is satisfied:(54)supz∈VhD′u;v,zz1,ΩW≥α1v1,ΩW,∀v∈Vh,where α1>0 is a constant independent of hh<h0.We define the Galerkin projection with respect toD′u;⋅,⋅ and Ph:V⟶Vh:(55)D′u;Phv,z=D′u;v,z,∀z∈Vh.Then, we obtain(56)v−Phv1,p,ΩW≤Cinfvh∈Vhv−vh1,p,ΩW≤Chσ,where 2≤p≤∞,0<σ<1.Lemma 5.
uhN∈Vh is a solution of (36) if and only if the following equation,(57)DN′uN;uN−uhN,v=RuN;uhN,v,∀v∈Vh,holds, where(58)RuN;uhN,v≜∫ΩW∫01∂2a∂s2x,whN∇whN⋅∇v1−tdtdhN2dx+2∫ΩW∫01∂a∂sx,whN∇dhN⋅∇v1−tdtdhNdx+∫0b∫0b∫01∂2a˜∂s2whN∂whN∂x2′∂v∂x2∑n=1N2nπsinnπx2′bsinnπx2b1−tdtdhN2dx2′dx2+2∫0b∫0b∫01∂a˜∂swhN∂dhN∂x2′∂v∂x2∑n=1N2nπsinnπx2′bsinnπx2b1−tdtdhNdx2′dx2,with whN=uN+tuhN−uN,dhN=uhN−uN.Proof.
Supposeηt≜DNwhN;whN,v. Then, by (32) and (36) and(59)η1=η0+η′0+∫01η'′t1−tdt.we can obtain the desired result.
Suppose(60)Mh≜v∈Vh|v1,∞,ΩW≤1+uN1,∞,ΩW.
Then, from [19, 25], we have the following lemma.Lemma 6.
There exists a constantC>0 independent of h, such that(61)RuN;v,z≤CuN−v1,ΩW2+uN−v1,ΩWz1,ΩW,∀v∈Mh,∀z∈Vh.We denote a nonlinear mappingψ:Vh⟶Vh, which satisfies that ψv is the unique solution of(62)D′u;ψv,z=D′u;u,z−Ru;v,z,∀z∈Vh,for any given v∈Vh. Suppose(63)Eh≜v∈Vh|v−Phv1,∞,ΩW≤Chσ.Then, we obtain the lemma as follows.Lemma 7.
The nonlinear mappingψ is a continuous mapping from Eh to Eh.Proof.
By (62), we have(64)D′u;ψv−ψvn,z=Ru;vn,z−Ru;v,z.
Combining (64) with (54), we deduce that the mapping ψ is continuous, i.e.,(65)limvn⟶vψvn=ψv.
For anyv∈Eh,(66)v1,∞,ΩW≤uN−v1,∞,ΩW+uN1,∞,ΩW,(67)uN−v1,∞,ΩW≤uN−PhuN1,∞,ΩW+PhuN−v1,∞,ΩW,(68)uN−PhuN1,∞,ΩW≤uN−ΠhuN1,∞,ΩW+ΠhuN−PhuN1,∞,ΩW.
SinceJh is regular and quasi-uniform, according to [26], we have the following inverse inequality:(69)w1,∞,ΩW≤Clog1h1/2w1,ΩW,∀w∈Vh.
By the definition ofEh, (56), and (69), we obtain(70)uN−v1,∞,ΩW≤1.
This implies thatv∈Mh. Under the definition of Ph, (62) can be rewritten as(71)D′uN;ψv−PhuN,z=−RuN;v,z,∀z∈Vh.
Then, by (54) and Lemmas 5 and 6, we obtain(72)ψv−PhuN1,ΩW≤Csupz∈VhD′u;ψv−PhuN,zz1,ΩW≤CuN−v1,ΩW2+uN−v1,ΩW≤CuN−PhuN1,ΩW2+PhuN−v1,ΩW2+uN−PhuN1,ΩW+PhuN−v1,ΩW≤Chσ.
This means thatψ:Eh⟶Eh.Theorem 3.
Supposeu∈V∩Wk,2+εΩW is a solution of problem (1), where ε>0,k≥2. We also assume that u|Γd0∈Hk−1/2Γd0 and u satisfies (53). With sufficiently small h, problem (36) has an approximate solution uhN∈Vh, such that(73)u−uhN1,ΩW≤Chσ+1N+1k−1ed0−dN+1π/buk−12,Γd0.Proof.
By Brouwer’s fixed-point theorem and Lemma7, there exists uhN∈Vh, such that ψuhN=uhN. From Lemma 5, we deduce that uhN is a solution of (36). Moreover, by (56) and uhN∈Eh, we have(74)uN−uhN1,ΩW≤uN−PhuN1,ΩW+PhuN−uhN1,ΩW≤Chσ,
For anyuN∈V, according to Lemma 3, we obtain(75)BuN;uN,v−BNuN;uN,v≤Ced0−dN+1π/b∑n=N+1+∞1+n21/2an21/2∑n=N+1+∞1+n21/2cn21/2≤C1N+1k−1ed0−dN+1π/b∑n=N+1+∞1+n2k−1/2an21/2∑n=N+1+∞1+n21/2cn21/2≤C1N+1k−1ed0−dN+1π/buk−1/2,Γd0v1,ΩW.
From (32), we have(76)DuN;uN,v=AuN;uN,v+BuN;uN,v=Fv+BuN;uN,v−BNuN;uN,v.
Letηt=Du+tuN−u;u+tuN−u,v; then,(77)∫01D′u+tuN−u;uN−u,vdt=DuN;uN,v−Du;u,v.
By (18), (52), and (53) and [24], we have(78)u−uN1,ΩW≤Csupv∈V∫01D′u+tuN−u;uN−u,vdtv1,ΩW≤CBuN;uN,v−BNuN;uN,vv1,ΩW≤C1N+1k−1ed0−dN+1π/buk−1/2,Γd0.
Combining (74) with (78), we obtain(79)u−uhN1,ΩW≤u−uN1,ΩW+uN−uhN1,ΩW≤Chσ+1N+1k−1ed0−dN+1π/buk−1/2,Γd0.
This completes the proof.


## 4. Numerical Examples

In this section, we computed some numerical examples by the method developed in Sections2 and 3 to test the efficiency of the method.Example 1.
We takeΩ=x1,x2|x1>0,0<x2<b,ΓW=0,x2|0<x2<b, ΓN=x1,b|x1>0, ΓS=x1,0|x1>0, b=1, and ax,u=1/1+u2. The exact solution of original problem is u=tan∑m=131/m2e−mπx1/bcosmπx2/b. Let ΓE=x1,x2|x1=d,0<x2<b be the artificial boundaries. Figure 3 shows the mesh h of subdomain ΩWd=1. The numerical results are given in Table 1 and Figures 4 and 5.
From the numerical results, we can deduce that the finite element mesh, the location of artificial boundary, and the truncation terms of series can affect the numerical errors. It is obvious that our method is very effective.Figure 3
Meshh of subdomain ΩW.Table 1
The errors with different meshesd=1,N=20.
MeshL2ΩW errorL∞ΩW errorh0.0278640.059187h/20.0077440.026333h/40.0020330.009699h/80.0005040.002785Figure 4
L∞ΓE errors with different Nd=1.Figure 5
L∞ΓE errors with different dN=20.


## 5. Conclusions

In this paper, we propose a method of artificial boundary conditions for quasi-linear problems in semi-infinite strips by using a segment artificial boundary. The exact and approximate artificial boundary conditions are given based on the Kirchhoff transformation. A new error estimate for the finite element approximation with the approximate artificial boundary condition is obtained. Finally, some numerical examples show the efficiency of this method. The quasi-linear problem, we considered in this paper, is a two-dimensional problem. Based on the proposed method, one can design some artificial boundary conditions for three-dimensional problem; we shall report on progress in some of these directions in a future publication.



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