Sheng Chen

School of Mathematics and Statistics,Jiangsu Normal University,Xuzhou 221116,China.

Abstract.Usual spectral methods are not effective for singularly perturbed problems and singular integral equations due to the boundary layer functions or weakly singular solutions.To overcome this difficulty,the enriched spectral-Galerkin methods(ESG)are applied to deal with a class of singularly perturbed problems and singular integral equations for which the form of leading singular solutions can be determined.In particular,for easily understanding the technique of ESG,the detail of the process are provided in solving singularly perturbed problems.Ample numerical examples verify the efficiency and accuracy of the enriched spectral Galerkin methods.

Key words:Singularly perturbed problems,weakly singular integral equations,boundary layers,enriched spectral Galerkin methods,Jacobi polynomials.

1 Introduction

As we know that the numerical error of the most existed numerical methods strictly rely on the regularity of the solutionuof the underlying problem.Especially,for the problems with high regularity solutions,spectral methods are capable of providing highly accurate solutions with significantly less unknowns than using a finite-element or finite difference methods[3,5,13,26].However,usual spectral methods based on orthogonal polynomials/functions do not have satisfactory convergence rate for singularly perturbed problems and singular integral equations due to solutions of those problems usually exhibited boundary layer phenomena or singular behaviours.More precisely,

·singularlyperturbedequations[7,11,15,17,23,28,34]:Given a tiny perturbed parametersε.The solutions of singularly perturbed equations usually involve the boundary layer,i.e.,the solutions change sharply(not tempered)in a narrow domain,such as,so that the usual spectral method cannot catch the information of the boundary layer functions accurately.

·Singularintegralequations[4,6,10,16,20,24,32,33]:The solutions of many integral equations with weakly singular kernel behave as a summation of Müntz polynomials,ri＞0(see[4,6]).Usually the index of the leading singular termr0is a small positive real number,so spectral methods are inefficient for singular integral equations due to the limited regularity of the solutions.The results of the spectral approximation to boundary layer functions behaving ascan be considerably improved by using special mapped polynomials[18,19,27,31],where singular mappings are used to establish spectral method with improved algebraic rates of convergence.However,these approximation results are still not uniform inε.Combing the parameterε,Schwab and Suri[23]use two-element spectral method(orpversion on two elements)to derive a robust exponential rate for boundary layer functions.Different from the singularly perturbed equations,the weak singularity of the solutions of the singular integral equations cannot derived the exponential convergence rate just by two elements due to the derivative of the solution are unbounded.So Wang[32]et al.and Yi[33]etal.useh-pfinite element methods,based on the geometric mesh[14],to handle Volterra integro-differential equations with smooth and weakly singular kernels.Meanwhile,by using a special mapping,Houetal.[16]derive an Müntz spectral method to enhance the convergence rate of the usual spectral method for singular integral equations.In this paper,we adopt the enriched spectral Galerkin method[9]to deal with several singularly perturbed problems and singular integral equations with a few boundary layer functions and leading singular terms which can be determined.The main merit of this method is that the enriched spectral Galerkin method keeps the structure of the usual spectral method.Especially in ESG-II,via a special property of the spectral method,we can obtain an improved numerical approximation just by repeating usual spectral method several times.

The remainder of this paper is organized as follows.In the next section,we provide a general framework of the enriched spectral-Galerkin methods.Moreover,we introduce the classical Jacobi polynomials and their basic properties which will be extensively used in subsequent sections.In Section 3,based on a Jacobi spectral Galerkin scheme and the analysis of the boundary layer functions,we apply ESG-II into several singularly perturbed problems.In Section 4,we study an singular integral equation,derive the form of the singular solutions and then apply ESG-II to obtain accurate solutions.Some concluding remarks are given in Section 5.

2 Preliminary

For the completeness of the statement,we provide a terse description of the enriched spectral Galekin method in the first part.Moreover,the definition and some basic properties of the Jacobi polynomials are listed in the second half of this section.

2.1 Enriched spectral Galerkin methods(ESG)

Spectral methods are capable of providing highly accurate solutions to smooth problems with significantly less unknowns than using a finite-element or finite difference methods[3,5,13,26].However,solutions to many singular perturbed/integral problems may deteriorate the convergence rate due to the weakly singular solutions or the boundary layer functions.In order to recover the high-accuracy of the spectral method for singularly perturbed problems and singular integral equations,we introduce the enriched Galerkin spectral methods in the regime of Galerkin method.More precisely,we consider the following weak formulation:Givenf∈X′,findu∈Xsuch that

whereXis a Hilbert space with norm‖·‖XandX′is its dual space,a(u,v)is a bilinear form inX×X.Letwithφn∈Xbeing certain smoothly orthogonal polynomials/functions such that the subspaceXN→X.Then,the classical spectral-Galerkin method is to finduN∈XNsuch that

Note that if the solutionuof the problem(2.1)is smooth and changed not so dramatically,then‖uN-u‖Xwill converge to zero rapidly.However,in many situations,the solutions of the singularly perturbed probelms and singular integral equations will not be smooth or tempered due to problems with weakly singular kernel in the integral operator or with a very small perturbed parameterε,so the traditional spectral methods with usual basis functions will not lead to accurate approximations.

2.1.1 Enriched spectral Galerkin method-I

For many singularly perturbed problems and singular integral equations,it is possible to determine the forms of a few leading singular terms or boundary layer functions.Assuming that thekfirst leading singular terms or boundary layer functions areψi,i=1,...,k,it is then natural to add those singular terms to the approximation spaceXN,leading to the so called enriched spectral method.Precisely,given a set of singular functionsψi,i=1,2,...,k,for the problem(2.1),we look fors.t.

Then the above system can be efficiently solved by forming the Schur-complement matrix CA-1B-D,and then we can obtainandsuccessively from

ESG-I is very efficient,and alleviates,to some extent,the ill conditioning problem caused by singular functions.However,the numerical results can still be plagued by ill conditioning askincreased,so we prefer to use the following ESG-II in practice.

2.1.2 Enriched spectral Galerkin method-II

A special feature of the spectral methods is that,for smooth functions,their spectral expansion coefficients will decay exponentially fast.Based on this property,we can establish ESG-II to solve many singular problems.For the detail of this new numerical method,one can refer to the recent work[9].Here comes a short description as follows.

Similar to ESG-I,we assume that a few leading singular terms of the underlying problems can be determined and denoted byψi,i=1,2,...,k.Here we only need to repeat the classical spectral scheme(2.2)several times instead of the enriched spectral scheme(2.3).For easily understanding ESG-II,we list some facts as follows:

·There exist some constantsc1,c2,...,cksuch thatu=u*+(c1ψ1+c2ψ2+...+ckψk),whereu*is the smooth part(compare withuandψi).The main work of the ESG-II is to derive the corresponding numerical approximationsci,N.

·Owing to the fact that the resource termfis known and the underlying singular termsψi,i=1,2,...,kcan be determined,we can use the spectral scheme(2.2)to derive the numerical solutions that

·We cannot derive the numerical solutionof the smooth partu*due to constantsci,i=1,2,...,kare unknown.However,we have the relation

Then,forn=1,2,...,N,the corresponding coefficients hold

·Without loss of generality,we can derive the numerical approximationsci,Nby setting*,n=0 forn=N-k+1,...,Ndue to the coefficients*,nof the smooth partu*converge to zero at a high rate of speed.Then we have a numerical approximation tou*below

The above facts exhibits that we have the numerical approximations for constantsciand the smooth partu*.Combing the singular termsψi,we obtain the numerical solution of ESG-II that

2.2 Jacobi polynomials

Letα,β＞-1 andωα,β(x)=(1-x)α(1+x)β.The Jacobi polynomialsare mutually orthogonal and satisfy

In particular,forα=β=0 andα=β=-1/2,Jacobi polynomialsare the Legendre polynomialsLn(x)and the Chebyshev polynomialsTn(x)up to a constant,respectively.

Jacobi polynomials are extensively investigated and there are abundant properties(see[1,5,13,25]and their references therein).Here we list some important properties as follows:

1.Sturm-Liouvilleproblem

where the singular Sturm-Liouville operator

2.Closedform

3.Three-termrecurrencerelation

4.Derivativerelation

where

In particular,in order to meet the homogeneous boundary condition arisen in many cases,we need to use a special generalized Jacobi polynomial

Similar to the classical Jacobi polynomials,it holds the following derivative relation

3 Applications to singularly perturbed equations

In this section,we will apply the ESG-II to the following singularly perturbed equations

whereε∈(0,1)and constantsb∈,c≥0.

The variational formulation of the singularly perturbed equations can be read as:Findsuch that

The well-posedness of the variational formulation can be derived by Lax-Milgram lemma and the following coercivity and continuity

Then,we can find a numerical solutionuNin the suitable finite dimensional subspace,i.e.,to findsatisfying

3.1 SPPs with one side boundary layer

We consider the following special caseb/=0,c=0 as the start point,

Note that the homogeneous linear equation-εu′′(x)+bu′(x)=0 has the fundamental solutionsc1+c2ebx/ε.We can see that the sign of the parameterbdetermine the location of the boundary layer.More precisely,forb＜0 the boundary layer will arises near the left end point;forb＞0 the boundary layer will arise near the right end point.

3.1.1 For b＜0

If the parameterb＜0,the boundary layer will arise near the left end point-1,and the boundary layer function behave asebx/ε.In order to meet the homogeneous boundary conditions,we take the singular term

We state the detail of the ESG-II as follows:

1.Substituting the singular termψinto the singularly perturbed problem(3.5),it’s straightforward to know thatψis the solution of the problem

Then we solve the above problem by the process(3.2)-(3.4)and derive the numerical solution

2.Givenf∈L2(I).Following the same process(3.2)-(3.4),the numerical approximationuNof the solutionu=u*+cψcan be derived that

whereu*,Nis the numerical approximation to the smooth partu*.By assuming the last one coefficient*,N=0,we have the numerical approximation

Note that the numerical solutionu*,Nis unknown,we just used the property that the coefficients*,ndecay to zero very fast due tou*is smooth.But we can approximate the smooth partu*by

3.Withψ,cNandin hands,we can install a new approximation toubelow

In order to test the validity of the ESG-II,we takef≡1-πcos(πx)+επ2sin(πx)andb=-1.Then the related exact solutionucan be detected that

With fixedN=35,we plot the error curves of theu-uN,ψN-ψand-uin Figure 1.The numerical results show that the ESG-II is effective comparing with the usual spectral method.Furthermore,to show the efficiency of the ESG-II,we draw the convergence curves in the left of Figure 2,in which the errors measured by discreteL2norm below

where{xj,ωj}are the related Gauss nodes and weights of the Legendre polynomialsLN(x).

Note that the numerical solutionsuNandψN,derived from the usual spectral method,are inefficient to approximate solutionsuandψdue to boundary layer functions changed sharply.However,we can derive the high-accuracy numerical solutionsfrom the low-efficiency numerical solutionsuNandψNvia the special techniques(3.7)-(3.11).In a sense,ESG-II can recovers the high efficiency for singularly perturbed equations just by repeating the usual spectral method several times.

Figure 1:Left:ε=10-3,N=35;Right:ε=10-6,N=35.

3.1.2 For b＞0

If the parameterb＞0,the boundary layer will arise near the right end point 1.We can follow the same process as the caseb＜0 but the singular term

and the related auxiliary problem

Then we can follow the process(3.7)-(3.11)to solve the singularly perturbed equation(3.5)as before.We take the solutionuandfas follows:

The numerical results plotted in Figure 2 illustrate that ESG-II is efficient for singularly perturbed equations withb＞0 and small parametersε.

Figure 2:Left:b=-1,ε=10-p;Right:b=1,ε=10-p.

3.2 SPPs with two sides boundary layer

We consider another interesting case:b=0,namely,the following singularly perturbed problem

whereε＞0 andc＞0.We can see from Schwab and Suri[23]that,for given smoothf,the solution of the problem(3.14)decomposed into a smooth partu*(x)and the boundary layer functions.To meet the boundary conditions,we take the following two boundary layer functions(BLF)and the related auxiliary problems(AP)

Repeating the spectral scheme(3.3)for approximatingu,ψ1andψ2,we can derive the related numerical approximationsuN,ψ1,Nandψ2,N,respectively.Obviously we can expand those numerical solutions as

Similar to the previous cases,we assume that the smooth partu*can be approximated by,thenuN=u*,N+c1ψ1+c2ψ2implies that

Without loss of generality,we can derive the approximation valuesc1,N,c2,Nof the constantsc1,c2by setting*,N-1=0 and*,N=0.Then we have the new approximation to the smooth partu*below

Finally,we obtain the numerical solution

Similar to the previous cases,for validating the numerical scheme,we first take

and the corresponding exact solution

The left of the graph demonstrates that the numerical solutions of the ESG-II exponentially converge to the solutions,where the parametersc=1 andε=10-p.

Figure 3:Left:c=1,ε=10-p;Right:c=1,ε=10-p.

Next,for givenf=excosπx,the right of Figure 3 shows that the numerical errors decay to zero exponentially,in which the exact solutions are replaced by the reference solutions,N=100 due to the exact solutions can not be detected.

4 Applications to singular integral equations

In this section,we are devoted to applying the enriched spectral method to deal with an integral equation with weakly singular kernelK(x,s):=(x-s)μ-1q(s)below

SpectralGalerkinmethod

Fora,b∈andρ∈+,the left and right fractional integrals are respectively defined as(see e.g.,[21,22]):

The weak formulation of the integral equation is to findu∈L2(I)such that

where the second term of the left hand side owes tofor allu,v∈L(I).Then we can approximate the weak solution in polynomials space.In order to derive the simple and well-conditioned matrix system,we propose the Legendre spectral scheme as follows:FinduN∈PN(I)such that

where we can expand the numerical solution

By substitutingv(x)=Lm(x),m=0,1,...,Nsuccessively into the scheme(4.4),we obtain the matrix system

Owe to the orthogonality of Legendre polynomials,it’s easy to know that M is a diagonal matrix with entries 2δnm/(2n+1).Moreover,via the relation[8,Lemma 2.4]

we can derive the matrix Iμby Jacobi Gauss quadrature formula with(α,β)=(μ,0).

EnrichedspectralGalerkinmethod

We can see from the previous section that the prerequisites of ESG-II are:i)an easily implemented spectral scheme;ii)the determined leading singular terms.With the spectral scheme(4.4),the remainder is to determine the singular terms.Fortunately,owe to the analysis of singularity in[4,6]and the mappingx=2t-1,t∈(0,1),we have a consequence of the solution’s singularity of the integral problem(4.1)below.

Then,using the enriched spectral Galerkin method withμ=0.7,M=10,it can be observed from the left of Figure 4 that the convergence rate can be enhanced by subtracting singular terms step by step.

Next,letq(x)=ex,μ=0.57.We consider a smooth functionf(x)=sin(x)as the data which can be approximated by polynomial efficiently as follows

wherefm∈Cm(),m→∞.According to the consequence of the Proposition 4.1,the solutionuhas the form

The right of Figure 4 depicts the convergence rate of the usual spectral method and the enriched spectral Galkerin method withk=1,2,3,where we deem,N=200 as the solutionudue to there is no exact solution can be detected.

5 Conclusion

In this work we consider several singularly perturbed problems and singular integral equations with a few boundary layer functions/leading singular terms which can be determined.In order to recovery the high-efficiency of the spectral method,the enriched spectral Galerkin methods(ESG),in the same spirit of extended or generalized finite element method[2,12,29],are applied to derive the accurate solutions.Successful implementations of ESG rely on three ingredients:(i)analyse the underlying problem and determine a few leading singular terms(ii)modify the singular functions to meet the boundary conditions of the underlying problem;(iii)use ESG-II,which is based on a special property of the spectral methods,to approximate the solution in the enriched spectral space.In particular,the detail of the process(iii)combing the singularly perturbed problems are provided in Section 3 for easily understanding the algorithm.Ample numerical examples showed that ESG-II is capable of producing significantly improved results over the usual spectral-Galerkin methods with adding only a few boundary layer/singular functions.