The linear singularly perturbed convection-diffusion problem in one dimension
is considered and its discretization on a Bakhvalov-type mesh is analyzed. The
preconditioning technique is used to obtain the pointwise convergence uniform
in the perturbation parameter.Comment: 12 page

Nhan, Thái Anh

Vulanović, Relja

English

arXiv

The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on a Bakhvalov-type mesh is analyzed. The preconditioning technique is used to obtain the pointwise convergence uniform in the perturbation parameter

Nhan, Thái Ahn

Scholar Commons - Santa Clara University

Santa Clara University Scholar Commons Mathematics and Computer Science College of Arts & Sciences 2015 Uniform convergence on a Bakhvalov-type mesh using the preconditioningapproach. Technical report Thái Ahn Nhan Santa Clara University, anhan@scu.edu Relja Vulanović Follow this and additional works at: https://scholarcommons.scu.edu/math_compsci  Part of the Mathematics Commons Recommended Citation Nhan, T. A., & Vulanović, R. (2015). Uniform convergence on a Bakhvalov-type mesh using the preconditioning approach. Technical report, arXiv:1504.0483. This technical report is a supplement to: Nhan, T. A., & Vulanović, R. (2016). Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes. Advances in Numerical Analysis, 2016, e2161279. This Other is brought to you for free and open access by the College of Arts & Sciences at Scholar Commons. It has been accepted for inclusion in Mathematics and Computer Science by an authorized administrator of Scholar Commons. For more information, please contact rscroggin@scu.edu. arXiv:1504.04283v1  [math.NA]  16 Apr 2015Uniform convergence on a Bakhvalov-type meshusing the preconditioning approach:Technical reportThái Anh Nhana,∗ and Relja VulanovićbaSchool of Mathematics, Statistics and Applied Mathematics,National University of Ireland, Galway, IrelandbDepartment of Mathematical Sciences, Kent State University at Stark,6000 Frank Ave. NW, North Canton, OH 44720, USAEmail: a.nhan1@nuigalway.ie, rvulanov@kent.eduApril 8, 2018AbstractThe linear singularly perturbed convection-diffusion problem in onedimension is considered and its discretization on a Bakhvalov-type meshis analyzed. The preconditioning technique is used to obtain the pointwiseconvergence uniform in the perturbation parameter.Keywords: singular perturbation, convection-diffusion, boundary-value prob-lem, Bakhvalov-type mesh, finite differences, uniform convergence, precondi-tioning2000 MSC: 65L10, 65L12, 65L20, 65L701 IntroductionThe report is a supplement to [8].2 The continuous problemWe consider the problemLu := −εu′′ − b(x)u′ + c(x)u = f(x), x ∈ (0, 1), u(0) = u(1) = 0, (1)with a small positive perturbation parameter ε and C1[0, 1]-functions b, c, andf , where b and c satisfyb(x) ≥ β > 0, c(x) ≥ 0 for x ∈ I := [0, 1].∗This author’s work is supported by the Irish Research Council under Grant No.RS/2011/179.1It is well known, see [3, 5] for instance, that (1) has a unique solution u inC3(I), which in general has a boundary layer near x = 0. Our goal is to findthis solution numerically.The solution u can be decomposed into the smooth and boundary-layer parts.We present here Linß’s [4, Theorem 3.48] version of such a decomposition:u(x) = s(x) + y(x), (2)|s(k)(x)| ≤ C(1 + ε2−k), |y(k)(x)| ≤ Cε−ke−βx/ε, (3)x ∈ I, k = 0, 1, 2, 3.Above and throughout the report, C denotes a generic positive constant whichis independent of ε. For the construction of the function s, see [4], since thedetails are not of interest here. As for y, it is important to note that it solvesthe problemLy(x) = 0, x ∈ (0, 1), y(0) = −s(0), y(1) = 0,with a homogeneous differential equation. We shall use this fact later on in thereport.3 The discrete problem and condition numberestimateWe first define a finite-difference discretization of the problem (1) on a generalmesh IN with mesh points xi, i = 0, 1, . . . , N , such that 0 = x0 < x1 < · · · <xN = 1. Throughout the rest of the paper, the constants C are also independentof N .Let hi = xi−xi−1, i = 1, 2, . . . , N , and ~i = (hi+hi+1)/2, i = 1, 2, . . . , N−1.Mesh functions on IN are denoted by WN , UN , etc. If g is a function definedon I, we write gi instead of g(xi) and gN for the corresponding mesh function.Any mesh functionWN is identified with an (N+1)-dimensional column vector,WN = [WN0 ,WN1 , . . . ,WNN ]T , and its maximum norm is given by∥∥WN∥∥ = max0≤i≤N|WNi |.For the matrix norm, which we also denote by ‖·‖, we take the norm subordinateto the above maximum vector norm.We discretize the problem (1) on IN using the upwind finite-differencescheme:UN0 = 0,LNUNi := −εD′′UNi − biD′UNi + ciUNi = fi, i = 1, 2, . . . , N − 1, (4)UNN = 0,whereD′′WNi =1~i(WNi+1 −WNihi+1− WNi −WNi−1hi)andD′WNi =WNi+1 −WNihi+1.2The linear system (4) can be written down in matrix form,ANUN = f̂N , (5)where AN = [aij ] is a tridiagonal matrix with a00 = 1 and aNN = 1 beingthe only nonzero elements in the 0th and Nth rows, respectively, and wheref̂N = [0, f1, f2, . . . , fN−1, 0]T .It is easy to see that AN is an L-matrix, i.e., aii > 0 and aij ≤ 0 if i 6= j, forall i, j = 0, 1, . . . , N . The matrix AN is also inverse monotone, which means thatit is non-singular and that A−1N ≥ 0 (inequalities involving matrices and vectorsshould be understood component-wise), and therefore an M -matrix (inversemonotone L-matrix). This can be proved using the following M -criterion, see[2] for instance.Theorem 1. Let A be an L-matrix and let there exist a vector w such thatw > 0 and Aw ≥ γ for some positive constant γ. A is then an M -matrix and itholds that ‖A−1‖ ≤ γ−1‖w‖.To see that AN is an M -matrix, just set wi = 2 − xi, i = 0, 1, . . . , N inTheorem 1 to get that ANw ≥ min{1, β}. This also implies that the discreteproblem (5) is stable uniformly in ε,‖A−1N ‖ ≤2min{1, β} ≤ C. (6)Of course, the system (5) has a unique solution UN .4 A Bakhvalov-type meshA generalization of the Bakhvalov mesh [1] to a class of Bakhvalov-type meshescan be found in [9]. Here we take one of the Bakhvalov-type meshes from [9] forthe discretization mesh IN . We refer to this mesh as as Vulanović-Bakhvalovmesh (VB-mesh). The points of the VB-mesh are generated by the function λin the sense that xi = λ(ti), where ti = i/N . The mesh-generating function λis defined as follows:λ(t) ={ψ(t), t ∈ [0, α],ψ(α) + ψ′(α)(t − α), t ∈ [α, 1],(7)with 0 < q < 1 and ψ = aεφ, whereφ(t) =tq − t =qq − t − 1, t ∈ [0, α].On the interval [α, 1], λ is the tangent line from the point (1, 1) to ψ, touchingψ at (α, ψ(α)). The point α can be determined from the equationψ(α) + ψ′(α)(1 − α) = 1.Since φ′(t) = q/(q − t)2, the above equation reduces to a quadratic one,aεα(q − α) + aεq(1− α) = (q − α)2,3which is easy to solve for α:α =q −√aεq(1− q + aε)1 + aε.We have to assume that aε < q (which is equivalent to ψ′(0) < 1) and thenα > 0. Note also that α < q andq − α = ζ√ε, ζ ≤ C, 1ζ≤ C. (8)Let J be the index such that tJ−1 < α ≤ tJ . Starting from the mesh point xJ ,the mesh is uniform, with step size H . However, xJ behaves differently fromthe transition point of the Shishkin mesh becausexJ ≥ ψ(α) =aαζ√ε.We note that the transition point ψ(α) is different also from the Bakhvalov-Shishkin of Vulanović-Shishkin meshes in the sense of [7].We now give the estimate for the condition number of AN when the dis-crete problem (4) is formed on the VB-mesh as described above. The conditionnumber isκ(AN ) := ‖A−1N ‖‖AN‖.We estimate the upper bound for ‖AN‖ by examining the entries of the matrixAN directly,‖AN‖ ≤ CN2ε.Combining this with (6), we get the following result.Theorem 2. The condition number of AN on the VB-mesh satisfies the follow-ing sharp bound:κ(AN ) ≤ CN2ε.5 ConditioningLet M = diag (m0,m1, . . . ,mN ) be a diagonal matrix with the entriesm0 = 1, mi =~iH, i = 1, 2, . . . , N − 1, and mN = 1.In other words,m0 = 1, mi =~iH, i = 1, 2, . . . , J, and mi = 1, i = J + 1, . . . , N. (9)When the system (5) is multiplied by M , this is equivalent to multiplying theequations 1, 2, . . . , J of the discrete problem (4) by ~i/H, i = 1, 2, . . . , J . Themodified system isÃNUN =Mf̃N , (10)4where ÃN =MAN . Let the entries of ÃN be denoted by ãij , the nonzero onesbeingli := ãi−1,i =− εhiH, 1 ≤ i ≤ J − 1,− εhJH, i = J,− εH2, J + 1 ≤ i ≤ N − 1,ri := ãi,i+1 =− εhi+1H− bi~ihi+1H, 1 ≤ i ≤ J − 1,− εH2− bi~iH2, i = J,− εH2− biH, J + 1 ≤ i ≤ N − 1,anddi := ãii =1, i = 0−li − ri +~iHci, 1 ≤ i ≤ J,−li − ri + ci, J + 1 ≤ i ≤ N − 1,1, i = N.Unlike the Shishkin mesh, which is piece-wise uniform, the VB-mesh isgraded in the fine part. Because of this, it is more difficulty to prove the uniformstability of the modified scheme. This is done in Lemma 2 below, but first weneed some crucial estimates for the graded mesh defined by (7).Lemma 1. For the mesh-generating function given in (7), the following esti-mates hold true:ε(hi+1 − hi)hihi+1≤ 2a, i = 1, 2, . . . , J − 2, (11)andε(H − hJ)hJH≤ ζ√εaq. (12)Proof. For i ≤ J − 2, we havehi = xi − xi−1 = aε(qq − ti− qq − ti−1)=aεqN(q − ti−1)(q − ti),hi+1 =aεqN(q − ti)(q − ti+1),andhi+1 − hi =2aεqN2(q − ti−1)(q − ti)(q − ti+1).5Then (11) follows becauseε(hi+1 − hi)hihi+1=2(q − ti)aq=2a(1− tiq)≤ 2a.The proof of (12) is more complicated due to the presence of hJ . First,hJ = γ1 + γ2, where γ1 = xα − xJ−1, γ2 = xJ − xα, and xα = ψ(α). Sinceγ2 = ψ′(α)(tJ − α)=aεqq − α(tJ − αq − α)andγ1 = aε (φ(α) − φ (tJ−1))= aε(αq − α −tJ−1q − tJ−1)=aεqq − α ·α− tJ−1q − tJ−1,we havehJ =aεqq − α[tJ − αq − α +α− tJ−1q − tJ−1]=aεq(q − α)2[tJ − α+(q − α)(α − tJ−1)q − tJ−1]=aεqζ2[tJ − α+ζ√ε(α− tJ−1)q − tJ−1].Moreover,ψ′(α) =aεq(q − α)2 and H = xJ+1 − xJ =ψ′(α)N,implying thatH =aεqN(q − α)2 .Therefore,H − hJ =aεqq − α[1N(q − α) −tJ − αq − α −α− tJ−1q − tJ−1]=aεqq − α[α− tJ−1q − α −α− tJ−1q − tJ−1]=aεqq − α (α− tJ−1)[1q − α −1q − tJ−1]=aεqq − α (α− tJ−1)α− tJ−1(q − α)(q − tJ−1)=aεq(q − α)2 ·(α− tJ−1)2q − tJ−1.6We now haveεH − hJhJH=aε2q(q − α)2 ·(α− tJ−1)2q − tJ−1· q − αaεq· 1tJ−αq−α +α−tJ−1q−tJ−1· (q − α)2Naεq=(q − α)Naq· (α− tJ−1)2q − tJ−1· (q − α)(q − tJ−1)qN − α2 + 2αtJ−1 − tJ−1tJ=(q − α)2Naq· (α − tJ−1)2ω≤ ζ2εaqN· 1ω,whereω :=qN− α2 + 2αtJ−1 − tJ−1tJand where in the last step we used (8) and the fact that 0 ≤ α − tJ−1 ≤ 1/N .The denominator ω can be estimated as follows:ω =qN− (α− tJ−1)2 −tJ−1N=ζ√ε+ αN− (α− tJ−1)2 −tJ−1N=ζ√εN+1N(α− tJ−1)− (α− tJ−1)2=ζ√εN+ (α− tJ−1) (tJ − α)≥ ζ√εN, since (α− tJ−1) (tJ − α) ≥ 0.Therefore,εH − hJhJH≤ ζ2εaqN· Nζ√ε=ζ√εaq.This completes the proof of (12).It is easy to see that ÃN is an L-matrix. The next lemma shows that ÃN isan M -matrix and that the modified discretization (10) is stable uniformly in ε.Lemma 2. Let ε be sufficiently small, independently of N , and let a > 4/β.Then the matrix ÃN of the system (10) satisfies∥∥∥Ã−1N∥∥∥≤ C.Proof. We want to construct a vector v = [v0, v1, . . . , vN ]T such that(a) vi ≥ δ, i = 0, 1, . . . , N , where δ is a positive constant independent of bothε and N ,(b) vi ≤ C, i = 0, 1, . . . , N ,(c) σi := livi−1 + divi + rivi+1 ≥ δ, i = 1, 2, . . . , N − 1.Then, according to the M -criterion,‖Ã−1N ‖ ≤ δ−1‖v‖ ≤ C.7The following choice of the vector v is motivated by [6, 11, 8]:vi =α−Hi+ λ, i ≤ J − 1,α−Hi+ λ1 + ρJ(1 + ρ)J−i, i ≥ J,where ρJ = βhJ/(2ε), ρ = βH/(2ε), and α and λ are fixed positive constants.Since HN ≤ C, there exists a constant α such that vi ≥ α−Hi ≥ δ > 0, so thecondition (a) is satisfied. Then, because of vi ≤ α + λ, the condition (b) holdstrue if we show that λ ≤ C. We do this next as we verify the condition (c).When 1 ≤ i ≤ J − 2, we use (11) to getσi = (li + di + ri)vi + liH − riH=~iHcivi −εhi+εhi+1+bi~ihi+1≥ −(εhi− εhi+1)+bi2+bihi2hi+1= −ε(hi+1 − hi)hihi+1+bi2+bihi2hi+1≥ −2a+bi2≥ β2− 2a=: δ > 0.The constant δ exists because of the assumption a > 4/β.For i = J − 1, we haveσJ−1 =~J−1HcJ−1vJ−1 + lJ−1H − rJ−1H+ λlJ−1 + λdJ−1 + rJ−1λ1 + ρJ≥ − εhJ−1+εhJ+bJ−1~J−1hJ− rJ−1λρJ1 + ρJ≥ − εhJ−1+bJ−12− rJ−1λρJ1 + ρJ≥ − εhJ−1+β2+(εhJH+bJ−1~J−1hJH)λβhJ2ε+ βhJ= − εhJ−1+β2+(2ε+ bJ−1(hJ−1 + hJ)2hJH)λβhJ2ε+ βhJ≥ β2− εhJ−1+λβ4H≥ β2> δwith a suitable positive constant λ. We can choose such λ because the estimatesH ≤ 2N−1 and q − tJ−1 ≤ q − tJ−2 ≤ 1 implyλβ4H− εhJ−1=λβ4H− Naq(q − tJ−1) (q − tJ−2) ≥ N(λβ8− 1aq)≥ 0.8For i = J , we getσJ =~JHcJvJ + lJH − rJH + λ[lJ +dJ1 + ρJ+rJ(1 + ρJ )(1 + ρ)]≥ −εhJ+εH+bJ~JH+λ(1 + ρJ )(1 + ρ)[lJ (1 + ρJ)(1 + ρ) + dJ(1 + ρ) + rJ ]≥εH−εhJ+bJ2+λ(1 + ρJ )(1 + ρ)[lJ (1 + ρJ)(1 + ρ) + dJ(1 + ρ) + rJ ]≥β2−ε(H − hJ )hJH≥ δ > 0.The above estimate holds true because (12) implies thatε(H − hJ)hJH≤ ζ√εaq≤ 2a,when ε is sufficiently small, and because we can show that[lJ(1 + ρJ )(1 + ρ) + dJ(1 + ρ) + rJ ] ≥ 0.Indeed,lJ(1 + ρJ)(1 + ρ) + dJ (1 + ρ) + rJ = lJρJ + lJρJρ− rJρ= − εhJHβhJ2ε− εhJHβhJ2εβH2ε+[εH2+bJ~JH2]βH2ε= −β24ε+βbJ~J2Hε= −β24ε+βbJ(hJ +H)4Hε≥ −β24ε+βbJ4ε≥ 0.Finally, when J + 1 ≤ i ≤ N − 1, we haveσi = civi + liH − riH +li1 + ρJ[λ(1 + ρ)i−1−J− λ(1 + ρ)i−J]+ri1 + ρJ[λ(1 + ρ)i+1−J− λ(1 + ρ)i−J]≥ bi +ρ(1 + ρ)li − ρri(1 + ρJ )(1 + ρ)i+1−Jλ≥ β2+(li − ri + liρ)ρ(1 + ρJ )(1 + ρ)i+1−Jλ=β2+(biH− β2H)λρ(1 + ρ)J−i−11 + ρJ≥ β2> δ.9By examining the elements of the matrix ÃN , we see that‖ÃN‖ ≤ CN2.When we combined this with Lemma 2, we get the following result.Theorem 3. The matrix ÃN of the system (10) satisfiesκ(ÃN ) ≤ CN2.6 Uniform convergenceLet τi, i = 1, 2, . . . , N − 1, be the consistency error of the finite-difference oper-ator LN ,τi = LNui − fi.We haveτi = τi[u] := LNui − (Lu)iand by Taylor’s expansion we get that|τi[u]| ≤ Chi+1(ε‖u′′′‖i + ‖u′′‖i), (13)where ‖g‖i := maxxi−1≤x≤xi+1 |g(x)| for any C(I)-function g. Let us defineτ̃i[u] =~iHτi[u], 1 ≤ i ≤ J,τi[u], J + 1 ≤ i ≤ N − 1.Lemma 3. The following estimate holds true for all i = 1, 2, . . . , N − 1:|τ̃i[u]| ≤ CN−1.Proof. We use the decomposition (2) and estimates (3). For the smooth part ofthe solution, it is easy to show that |τ̃ [s]| ≤ CN−1. Then we need to show that|τ̃i[y]| ≤ CN−1.Case 1. Let i ≥ J + 1, i.e. ti−1 ≥ tJ ≥ α. Then we have|τ̃i[y]| = |τi[y]| ≤ Chi+1 (ε‖y′′′‖i + ‖y′′‖i)≤ CN−1λ′(ti+1)ε−2e−βλ(ti−1)/ε≤ CN−1λ′(ti+1)ε−2e−βλ(α)/ε≤ CN−1ε−2e−aβα/(ζ√ε)≤ CN−1,where we have used the fact that ε−2e−aβα/(ζ√ε) ≤ C.10Case 2. Let i ≤ J , i.e. ti−1 < α, and at the same time, let ti−1 ≤ q − 3/N .Note that, when ti−1 ≤ q − 3/N , we haveti+1 ≤ q − 1/N < q and q − ti+1 ≥13(q − ti−1).This is becauseq − ti−1 ≥3N⇒ 23(q − ti−1) ≥2N,which givesq − ti+1 = q − ti−1 −2N=13(q − ti−1) +23(q − ti−1)−2N≥ 13(q − ti−1).Therefore,|τ̃i[y]| =~iH|τi[y]| ≤~iHChi+1 (ε‖y′′′‖i + ‖y′′‖i)≤ CN−1 [λ′(ti+1)]2 ε−2e−βλ(ti−1)/ε≤ CN−1 [φ′(ti+1)]2 e−aβφ(ti−1)≤ Cε−1N−1(q − ti+1)−4e−aβ(q/(q−ti−1)−1)≤ CN−1(q − ti−1)−4e−aβq/(q−ti−1)≤ CN−1,because (q − ti−1)−4e−aβq/(q−ti−1) ≤ C.Case 3. In the last case, we consider the remaining possibility, q − 3/N <ti−1 < α. We use the fact that Ly = 0 to work with|τ̃i[y]| =hiH|τi[y]| ≤~iH(Pi +Qi +Ri) ,wherePi = ε|D′′yi|, Qi = bi|D′yi|, and Ri = ci|yi|.We now follow closely the technique in [10, Lemma 5], (see also [11, 8]), to get~iH(Pi +Qi + Ri) ≤ C[~iH(1~iε · 2‖y′‖i)+~iH(1hi+1‖y‖i)+ e−βλ(ti)/ε]≤ CNe−βλ(ti−1)/ε≤ CNe−aβφ(ti−1)≤ CNe−aβφ(q−3/N)≤ CNe−aβ(qN/3−1)≤ CN−1.Remark 1. The technique used in the above proof is based on [9], wherethe same approach is successfully applied to reaction-diffusion problems. Thisapproach is originally due to Bakhvalov [1]. The technique works here forconvection-diffusion problems (1) because an extra ε-factor is obtained fromthe preconditioner (9).11When Lemmas 2 and 3 are combined, which amounts to the use of theconsistency-stability principle, we obtain the following result.Theorem 4. Let ε be sufficiently small, independently of N , and let a > 4/β.Then the solution UN of the discrete problem (5) on the VB-mesh satisfies∥∥UN − uN∥∥ ≤ CN−1,where u is the solution of the continuous problem (1).References[1] N. S. Bakhvalov, The optimization of methods of solving boundary valueproblems with a boundary layer, USSR Comp. Math. Math. Phys. 9 (1969)139-166.[2] E. Bohl, Finite Modelle gewöhnlicher Randwertaufgaben, Teubner, Stutt-gart, 1981.[3] R. B. Kellog, A. Tsan, Analysis of some difference approximations for asingular perturbation problem without turning points, Math. Comput. 32(1978) 1025–1039.[4] T. Linß, Layer-Adapted Meshes for Reaction-Convection-Diffusion Prob-lems, Lecture Notes in Mathematics, Vol. 1985, Springer, Berlin, Heidel-berg, 2010.[5] J. Lorenz, Stability and monotonicity properties of stiff quasilinear bound-ary value problems, Univ. u Novom Sadu Zb. Rad. Prir. Mat. Fak. Ser.Mat. 12 (1982) 151–175.[6] H.-G. Roos, A note on the conditioning of upwind schemes on Shishkinmeshes, IMA J. Numer. Anal. 16 (1996) 529–538.[7] H.-G. Roos, T. Linß, Sufficient condition for uniform convergence on layer-adapted grids, Computing, 63(1): 27-45, 1999.[8] T. A. Nhan, R. Vulanović, Preconditioning and uniform convergence forconvection-diffusion problems discretized on Shishkin-type meshes, 2015.[9] R. Vulanović, On a numerical solution of a type of singularly perturbedboundary value problem by using a special discretization mesh, Univ. uNovom Sadu Zb. Rad. Prir. Mat. Fak. Ser. Mat. 13 (1983) 187–201.[10] R. Vulanović, A priori meshes for singularly perturbed quasilinear two-point boundary value problems, IMA J. Numer. Anal. 21 (2001) 349–366.[11] R. Vulanović, T. A. Nhan, Uniform convergence via preconditioning, Int.J. Numer. Ana. Model. Ser. B 5, 347–356.12

Uniform convergence on a Bakhvalov-type mesh using the preconditioning approach: Technical report

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