Parabolic Layers, II

Transcription

1 Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers, II Paul A. Farrell, Pieter W. Hemker, and Grigori I. Shishkin Computer Science Program Technical Report Number CS March 10,1995 Department of Mathematics and Computer Science Kent State University Kent, Ohio

2 Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers, II æ P.A. Farrell y P.W. Hemker z G.I. Shishkin x 1 2 Abstract In his series of three papers we study singularly perturbed èspè boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the ænite diæerence scheme and the approximation of the diæusive æux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct diæerence schemes that allow approximation of the solution and the normalised diffusive æux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diæusion equations. Also for these problems we construct special ænite diæerence schemes, the solution of which converges "-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we ærst give anintroduction on the general problem, and then we consider respectively èiè Problems for SP parabolic equations, for which the solution and the normalised diæusive æuxes are required; èiiè Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; èiiiè Problems for SP parabolic equation with discontinuous boundary conditions. æ Note: This work was supported by NWO through grant IB y Department of Mathematics and Computer Science, Kent State University, Kent, OH 44240, U.S.A. z CWI, Center for Mathematics and Computer Science, Amsterdam, The Netherlands x IMM, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Science, Ekaterinburg, Russia 1 AMS Classiæcation: Primary: 65L12. Secondary: 65L50, 65N06, 65N50. 2 Keywords: singular perturbation problems, æ-uniform methods, parabolic boundary layers 1

3 Part II Boundary value problem for elliptic equation with mixed boundary condition 1 Introduction In this part we sketch avariety of special methods which are used for constructing "-uniformly convergent schemes. We shall demonstrate a method which achieves improved accuracy for solving singularly perturbed boundary value problem for elliptic equations with parabolic boundary layers. In Section 4 we shall introduce a natural class, B, of ænite diæerence schemes, in which èby the above mentioned approaches èaè and èbèè we can construct èformallyè the special ænite diæerence schemes with approximate solutions which converge parameter-uniformly to the solution of our initial boundary value problem. In this chapter we consider a class of singularly perturbed boundary value problems which arise when diæusion processes in a moving medium are modeled. For such boundary value problems which describe transfer with diæusion, we construct a special scheme that converges parameter-uniformly. We shall show that for the construction of such schemes from class B, the use of a special condensing grid èor an adaptive meshè is necessary. It means that the choice èto construct special parameter-uniformly convergent schemes for our class of convection diæusion problemsè is quite restricted. By condensing èor adaptiveè grids we can construct ænite diæerence schemes which converge parameter-uniformly. We shall present and discuss the results of numerical computations using both the classical and the new special ænite diæerence schemes. 2 The class of boundary value problems 2.1 The physical problem The diæusion of a substance in a convective æow of an incompressible æuid in a two-dimension domain gives rise to an equation of the form, "æuèxè+~vèxèæ ~ ruèxè=fèxè; x2æ; è2:1aè where ~vèxè and F èxè are the velocity and source, respectively; 1=" is the Peclet number èreynolds numberè, if the substance is heat èdiæusive matter or momentumè ë1ë. When the substance is heat èdiæusive matter or momentumè then 2

4 uèxè is the temperature èdensity or velocityè at the point x. On the boundary of domain considered èthat is the wall of the container holding the æuidè we have a boundary condition that describes the exchange of the substance with the surrounding environment, uèxè =0; æ: è2:1bè æ is the boundary of the domain èthe wall of the containerè, the outward normal derivative at the boundary, æ characterises intensity of exchange of the substance between the medium and the wall, where the value is given by Uèxè. When æ tends to inænity the condition è2.1bè becomes the Dirichlet condition uèxè =Uèxè; 0 æ: The inæow boundary, that is the part of the 0 æ where the stream enters the domain, we denote + æ, and the outæow boundary æ ~nèxè æ ~vèxè é 0; + æ; ~nèxè æ ~vèxè é 0; æ: Here ~nèxè is a unit vector in the direction of the external normal. 0 æwehave condition On the ~nèxè æ ~vèxè =0; 0 æ: + æ the value of uèxè is given, and on outæow æwe assume the æux to be known uèxè =U + èxè; + uèxè =æèxè; æ: è2:1dè Problem è2.1è describes a general diæusion process into moving medium. For suæciently large Peclet number èreynolds numberè, " can be very small. As " tends to zero a boundary layer appears in the neighbourhood of the 0 æ. 2.2 The class of boundary value problems Now we describe the class of two-dimensional convection-diæusion problems with mixed boundary conditions, for which we shall study the convergence behaviour. Notice that we consider here mixed boundary conditions, where usually only Dirichlet boundary conditions are studied. On the rectangular domain D = fx : 0 éx i éd i ; i=1;2gwe consider the elliptic boundary value problem L è2:2è uèxè ç è " 2 X s=1;2 a s 2 1 è uèxè =fèxè; x2è2.2aè D; 3

5 uèxè ='èxè; x2, + ; l è2:2è uèxè, è1, æèuèxè =èèxè; uèxè =çèxè; x2,, : è2.2bè è2.2cè è2.2dè Here a s ;b;c;f;';è, and ç are suæciently smooth functions, æ 2 ë0; 1ë, " 2 è0; 1ë, a 0 ç a 1 èxè, a 2 èxè ç a 0 ; bèxè ç b 0 ; cèxè ç 0; x 2 D; a 0 ; b 0 é0, and, + =,ëfxjx 1 =0g;,, =,ëfxjx 1 =d 1 g;, 0 =,ëfxj0éx 1 éd 1 g: This class of problems includes, for example, the following boundary value problem for a regular diæerential equation X L è2:3è Uèyè çf s=1;2 A 2 s y 2 e D, with regular boundary 1 guèyè =Fèyè; è2:3aè Uèyè =æèyè; y2e, + ; l è2:3è Uèyè Uèyè, è1, =æèyè; y2, e 0 ; =0; y2, e, ; on the rectangular domain e D = fy : 0 éyi ée di ; i=1;2g,e di =",1 d i, if the size of domain e D is suæciently large. 2.3 The construction of "-uniformly convergent schemes When " tends to zero in the neighbourhood of, 0, boundary layers appear which are described by parabolic equations. Hence these layers are known as parabolic boundary layers. Although classical diæerence approximations èsee, for example, ë6, 7ë ë è converge for è2.2è to the solution of the boundary value problem for each æxed value of " èsee Theorem 3.1è, the accuracy of the numerical solution depends on the value of " and decreases, sometimes to complete loss of accuracy, when " is less or comparable with the step-size of the uniform grid. This means that classical ænite diæerence schemes do not converge uniformly with respect to the parameter ", èsee theorem 3.2è. Therefore, for the boundary value problem è2.2è it is of interest to construct special schemes the solution of which does converge "-uniformly. For the case of the Dirichlet problem è2.2aè, an "-uniformly convergent ænite diæerence scheme is found in ë9, 10ë. 4

6 3 Classical diæerence scheme To solve the problem è2.2è we ærst use a classical ænite diæerence method. On the set D we introduce the rectangular grid D h =! 1 æ! 2 ; è3:4è where! s is a, in general non-uniform, grid on the interval ë0;d s ë and N s is the number of nodes of the grid! s, s =1;2. Deæne h i s = x i+1 s, x i s, h s = max i h i s, h ç MN,1, where h = max s h s, N = min s N s, s =1;2; D h = D ë D h,, h =,ëd h :For problem è2.2è we use the diæerence scheme where æ è3:5è zèxè ç " 2 æ è3:5è zèxè =fèxè; x2 D h ; zèxè ='èxè; x2, + h ; ç è3:5è zèxè =èèxè; x2, 0 h ; æ x1 zèxè =çèxè; x2,, h ; X s=1;2 a s èxèæ xsbx s zèxè, bèxèæ x1 zèxè, cèxèzèxè; è3:5aè è3:5bè è3:5cè è3:5dè ç è3:5è zèxè ç è "ææx2 zèxè, è1, æèzèxè; x 2 =0;,"ææ x2 zèxè, è1, æèzèxè; x 2 = d 2 ; æ xsbx s zèxè is the second divided diæerence on a non-uniform grid, and æ xs zèxè and æ xs zèxè are the ærst forward and backward divided diæerences. The diæerence scheme è3.5è, è3.4è is monotone èthat is the maximum principle holdsè ë7ë. By means of the maximum principle, and using the estimates of the derivatives èsee ë5ëè, we ænd that the solution of the scheme è3.5è-è3.4è converges èfor a æxed value of the parameter "è as juèxè,zèxèjçm",4 N,1 ; x 2 D h : è3:6è Theorem 3.1 Let u 2 C 4 èdè. Then, for a æxed value of the parameter ", the solution of the scheme è3.5èíè3.4è converges to the solution of the boundary value problem è2.2è with an error bound given by è3.6è. Clearly è3.6è does not imply "-uniform convergence of the diæerence scheme. In fact it can be shown that it is impossible to obtain "-uniform convergence for the diæerence scheme è3.5è-è3.4è on a æxed "-independent mesh. The proof is found in ë8ë. We summarise this result in the following theorem. Theorem 3.2 èsee ë5ëè On an "-independent grid of type è3.4è, the solution of the classical ænite diæerence scheme è3.5è,è4.8è does not converge "-uniformly to the solution of the boundary value problem è2.2è. 5

7 We want to make the following interesting observation. We consider problem è2.2è. If we take inl è2:2è the parameter " = 1 èleaving " unchanged in è2.2aè and æ è3:5è, but adapting it in ç è3:5è è, then no singular part will appear as a ærst term in the expansion w.r.t. ". Hence, the classical scheme will be "-uniform convergent in this case. 4 The ætted diæerence scheme For parabolic problems with parabolic layers, it was shown in ë5ë that there does not exist a diæerence scheme only based on ætting of the coeæcients, for which the solution converges "-uniformly to the solution. Here we show a similar result for the elliptic boundary value problem è2.2è. Let us consider the problem L è4:7è uèxè ç " 1 uèxè =0; x2d; è4:7aè uèxè =0; x2, + ; è4:7bè l è2:2è uèxè =èèxè; x2, 0 ; è4:7cè where uèxè =0; x2,, ; è4:7dè ç è0 èx 1 è; x2, 0 ; x 2 =0; 0; x2, 0 ; x 2 6=0; and the function è 0 èx 1 è; x 1 2 ë0;d 1 ë is suæciently smooth. The solution of problem è4.7è is the singular solution. Let us introduce a class, called class A, of ænite diæerence schemes for problem è4.7è, for the construction of which we use uniform meshes: D u h = fd hè3:4è ; where! s =! u s are uniform grids; s =1;2g: è4:8è and also èfor the approximation of equation è4.7aèè a standard æve-point, ætted ænite diæerence operator X æ è4:9è zèxè çf èa s æ xs x s + B s æ xs, Cgzèxè =E; x 2 D h : è4:9è s=1;2 Here the coeæcients A s ;B s ;C;Eare functionals of the coeæcients of equation è4.7aè and also depend on x; h 1 ; h 2, and ". We suppose that for h 2 ",1! 0 and h 1! 0 these coeæcients A s ; B s ; C; Eapproximate the data of equation è4.7aè, in the uniform norm, in the neighbourhood of at least one point the boundary layer region. 6

8 Theorem 4.1 In the class A of ænite diæerence schemes there does not exist a diæerence scheme of which the solution converges "-uniformly to the solution of the boundary value problem è4.7è. The proof of this theorem is rather complex. An outline of the principle steps is found in ë11, 13ë. Remark 1. A statement similar to Theorem 4.1 is also true in the case when the diæerence schemes are constructed on a more general stencil with a ænite number of nodes. The results of Theorem 4.1 and Remark 1 can be explained as follows. All solutions of problem è4.7è èdeæned by diæerent functions èèxèè are singular solutions. Those solutions can not be represented as linear combinations of a ænite number of æxed functions of boundary layer type èboundary layer functionsè. Let us introduce class B of ænite diæerence schemes for problem è2.2è, for the construction of which we use rectangular grids D h è3:4è, which are generally nonuniform and a æve-point ænite diæerence operator èin general a ætted operatorè of the standard form. The coeæcients of the diæerence operator are, as before, functionals of the coeæcients of the equation è2.2aè and also depend on x; " and on the distance between the nodes of the stencil used. Again, we suppose that for h! 0 the coeæcients of the diæerence operator approximate èin the uniform normè the coeæcients of equation è2.2aè on the set D h è3:4è. We remark that class B is a natural class for constructing ænite diæerence schemes for problem è2.2è as it includes both ætted methods and methods with special condensing grids. This is in contrast to class A, which contains only schemes on uniform meshes. Consequences of Theorem 4.1 include results such as: Corollary 4.2 In the case of boundary value problems of type è2.2è, class B of ænite diæerence schemes does not contain any diæerence scheme which, on grids with arbitrary distribution of nodes, can achieve "-uniform convergence of the solution to solution of boundary value problem è2.2è by the use of a ætted method. Corollary 4.3 In the case of boundary value problems of type è2.2è, the use of special condensing grids èor adaptive meshesè is necessary for the construction of " -uniformly class B ænite diæerence schemes. 5 Diæerence scheme of method of special condensing mesh We now construct an "-uniformly convergent scheme for the boundary value problem è2.2è. We use a special condensing mesh èin the neighbourhood of 7

9 the boundary layersè, where the distribution of the nodes is deæned by a-priori estimates of the solution and its derivatives. This approach is similar to that in ë11, 12, 13ë, where the Dirichlet problem was studied. Consider the special grid D æ h =! 1 æ! æ 2 ; è5:10è where! æ 2 =! æ 2 èçè is a special piecewise uniform mesh,! 1 is a uniform mesh, ç is a parameter which depends on " and N 2. The mesh! æ 2 èçè is constructed as follows. The interval ë 0;d 2 ë is divided into three parts ë 0;çë, ë ç;d 2,çë, ë d 2,ç;d 2 ë, 0 éççd 2 =4. Each subinterval ë 0;çë and ë d 2,ç;d 2 ë is divided into N 2 =4 equal cells and the subinterval ë ç;d 2,çëinto N 2 =2 equal cells. Suppose ç = çè"; N 2 è = minë d 2 =4; m"ln N 2 ë where m is arbitrary number. The diæerence scheme è3.5è, è5.10è belongs to class B. The scheme is constructed using an a priori adapted mesh. Distribution of the nodes on grid D æ h ensures " -uniform approximation of the boundary value problem. This is formalised in the following theorem èsee also ë5ëè. Theorem 5.1 The solution of diæerence scheme è3.5è, è5.10è converges " - uniformly to the solution of boundary value problem è2.2è. The following bound holds for the error juèxè, zèxèj çmn,1=3 ; x2d æ h : è5:11è The proof of this theorem will appear in a future paper. 6 Numerical results Theoretically èsee Theorem 3.2è it has been shown that the classical diæerence scheme è3.5è on the uniform grid è4.8è does not converge "-uniformly in the l 1 -norm to the solution of the boundary value problem è2.2è. But it could be the case that the error max Dh jzèxè, uèxèj is relatively small for the classical scheme, which would reduce the need for a special scheme. On the other hand, Theorem 5.1 shows that the special scheme è3.5è,è5.10è converges "-uniformly, but no indication is given about the value of the order constant M in è5.11è and the order of convergence is rather small. It might be that the error is relatively large for any reasonable values of N 1, N 2. This would reduce the practical value of the special scheme. The following numerical experiments address these issues. 6.1 The model problem To see the eæect of the special scheme in practice, for the approximation of the model problem we study the singularly perturbed elliptic equation with a mixed 8

10 boundary condition L è6:12è uèxè ç " 2 æuèxè, l è6:12è uèxè =èèxè; x2, 1 uèxè =,1; x2d ; è6:12è where uèxè=0; x2, 2 uèxè =0; x2,, ; l è6:12è uèxè ç ç æ"è@=@x2 èuèxè,è1, æèuèxè; x 2 =0;,æ"è@=@x 2 èuèxè,è1, æèuèxè; x 2 =1: We compare the numerical results for the classical scheme è3.5è, è4.8è and the special scheme è3.5è, è5.10è. Here D = fx :0éx 1 ;x 2 é1g, ç x1 ; x2, èèxè= 0 ;x 2 =0; 0; x2, 0 ;x 2 =1: For the solution of problem è6.12è, we have the representation uèxè =Uèxè+Wèxè; x2d; where Uèxè =x 1,x2D, is the outer solution, and W èxè represents the parabolic boundary layer in the neighbourhood of the edges at x 2 = 0 and x 2 =1. We have the following bounds on the solution,1 ç uèxè ç 1; x2d : Due to Theorem 5.1 the solution of the discrete problem with the adapted mesh converges "-uniformly to the solution of our model problem è6.12è. The function u h èxè, which is the solution of the special scheme è3.5è,è5.10è is shown in Figure The behaviour of the numerical solution of the classical scheme To see the diæerence between the use of the uniform and the adapted grid, for the approximation of è6.12è we ærst use the classical scheme è3.5è,è4.8è. We solve the problem for diæerent values of the mesh width h 1 = h 2 = N,1 and for diæerent values of the parameters " and æ. The results for a set of numerical experiments is given in Table 1. From Table 1 we can see that the solution of scheme è3.5è-è4.8è does not converge "-uniformly. The errors, for a æxed value of N, depend on the parameters " and æ. For " ç 0:1 and æ =0:0;0:1;0:5;1:0 the error behaviour is regular: when N increases, the error decreases. For " =10,3 9

11 Figure 1: Solution computed with the adapted mesh, " = 0:01, æ = 0:5, N = 32 and m =1:0. Table 1: Table of errors EèN; "; æè for the classical scheme N " æ è-1è 0.329è-1è 0.158è-1è 0.696è-2è 0.241è-2è 10, è-1è 0.291è-1è 0.122è-1è 0.403è-2è 10, è-1è è-1è 0.246è-1è 10, è-1è 0.283è-1è 0.157è-1è 0.200è-1è è-1è 0.381è-1è 0.170è-1è 0.744è-2è 0.314è-2è 0.104è-2è 10, è-1è 0.310è-1è 0.129è-1è 0.423è-2è 10, è-1è 10, è-1è 0.401è-1è 0.183è-1è 0.842è-2è 0.359è-2è 0.119è-2è 10, è-1è 0.176è-1è 10, , è-1è 0.288è-1è 0.132è-1è 0.562è-2è 0.187è-2è 10, è-1è 0.211è-1è 10, , In this table the error EèN; "; æè is deæned by EèN; "; æè = max x2d h jeèx; N; "; æèj ; è6:13aè eèx; N; "; æè =zèxè,u? èxè; è6:13bè where u? èxè is the piecewise interpolation of z m?256 èxè, m = m è5:10è = 1 èsee Table 2è, and zèxè ç z N èxè is the solution of è3.5è,è4.8è with h 1 = h 2 = N,1. Notice that u? èxè is an accurate approximation of uèxè. 10

12 and æ =0:0;0:1;0:5 and for " =10,2 and æ =0:0, for some values of N the error increases with increasing N. For æ =0:5;1:0 and a æxed N the error increases with decreasing ". In particular, for " = 10,3 and æ = 0:5;1:0 the errors for N ç 128 are of the same order or larger than èin `1-normè the solution of the BVP. Thus, the numerical results illustrate that the lack of"-uniform convergence leads to large errors indeed. 6.3 The behaviour of the numerical solution of the special scheme In Table 2 we show the behaviour of è3.5è,è5.10è, with m = m è5:10è = 1, applied to the model problem è6.12è From Table 2 we can see that the solution of the Table 2: Table of errors EèN; "; æè for the special scheme N " æ è-1è 0.329è-1è 0.158è-1è 0.696è-2è 0.241è-2è 10, è-1è 0.291è-1è 0.122è-1è 0.403è-2è 10, è-1è 0.361è-1è 0.148è-1è 0.497è-2è 10, è-1è 0.361è-1è 0.148è-1è 0.497è-2è è-1è 0.381è-1è 0.170è-1è 0.744è-2è 0.314è-2è 0.104è-2è 10, è-1è 0.310è-1è 0.129è-1è 0.423è-2è 10, è-1è 0.461è-1è 0.240è-1è 0.924è-2è 10, è-1è 0.461è-1è 0.241è-1è 0.925è-2è è-1è 0.401è-1è 0.183è-1è 0.842è-2è 0.359è-2è 0.119è-2è 10, è-1è 0.176è-1è 10, è-1è 10, è-1è è-1è 0.288è-1è 0.132è-1è 0.562è-2è 0.187è-2è 10, è-1è 0.211è-1è 10, è-1è 10, è-1è In this table the function EèN; "; æè is deæned by è6.13è, but now zèxè =z m?n èxè in è6.13è is the solution of è3.5è,è5.10è with m = m è5:10è = 1 and N 1 = N 2 = N scheme è3.5è-è5.10è does converge "-uniformly indeed. The errors for a æxed value of " =1:0;10,1 ; 10,2 ; 10,3 and æ =0:0;0:1;0:5;1:0have all a regular behaviour and decrease for increasing N. For a æxed value of æ and N the error stabilises for decreasing ": the errors for " =10,2 and " =10,3 are practically the same. For " ç 10,2 and a æxed value of N we ænd the largest error for æ =1:0. In particular, for " ç 10,2, æ =1:0 and N = 128 the error is less than 6è. Also here, the numerical results illustrate the practical value of "-convergent methods. 11

13 Conclusion For the elliptic boundary value problem è2.2è, where a small parameter multiplies the highest derivative, we have analysed diæerent approaches for the construction of discrete methods. We present methods for which the accuracy of the discrete solution does not depend on the value of the small parameter, but only on the number ofpoints in the discretisation. We show that in a natural class of ænite diæerence schemes, for the problem considered, no "-uniform methods exist on a uniform grid ètheorem 4.1è. As a consequence, for the construction of "-uniform methods the use of an adapted non-uniform mesh is necessary. With a special, adapted, non-uniform mesh and a simple classical diæerence scheme, we are able to construct an "-uniform approximation To illustrate the practical importance of our study, for a model problem we show byanumerical example that, on a uniform grid, the classical diæerence scheme is not "-uniformly convergent. In our example, the error èwith a Neumann boundary conditionè is not less than 700è of the solution, for N=128 and " =10,3. The same example shows that we might obtain an "-uniformly convergent solution if we use the adapted mesh. Now the error is not larger than 6è of the solution, for any value of the parameter ". Thus, the numerical example illustrates that the theoretical considerations have practical implications indeed. References ë1ë E.V. Ametistov V.A. Grigor'ev B.T. Eæmtsev, editor. Heat and Mass Transfer. HeatíTechnology Experiment. Handbook. Energoizdat, Moscow, èin Russianè. ë2ë P.W. Hemker and G.I. Shishkin. Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition. In J.J.H. Miller, editor, BAIL VI Proceedings, pages 3í4. Boole Press, ë3ë P.W. Hemker and G.I. Shishkin. Approximation of parabolic PDEs with a discontinuous initial condition. East-West Journal of Numerical mathematics, 1è4è:287í302, ë4ë P.W. Hemker and G.I. Shishkin. Discrete approximation of singularly perturbed parabolic pdes with a discontinuous initial condition. Computational Fluid Dynamics Journal, 2:375í392, ë5ë P.W. Hemker and G.I. Shishkin. On a class of singularly perturbed boundary value problems for which an adaptive mesh technique is necessary. In D. Bainov and V. Covachev, editors, Proceedings of the Second International Colloquium on Numerical Analysis, pages 83í92. VSP, International 12

14 Science Publishers, Proceedings of the ësecond International Colloquium on Numerical Analysis", August 13-17, ë6ë G.I. Marchuk. Methods of Numerical Mathematics. Springer, New York, ë7ë A.A. Samarski. Theory of Diæerence Scheme. Nauka, Moscow, èin Russianè. ë8ë G.I. Shishkin. An increase in accuracy of solutions to diæerence scheme for parabolic equations with small parameters at highest-order derivatives. Zh. Vychisl. Mat. Mat. Fiz., 24:864í875, èin Russianè. ë9ë G.I. Shishkin. Approximation of solutions to singularly perturbed boundary value problems with corner boundary layers. Doklady Akad. Nauk SSSR., 296:39í43, èin Russianè. ë10ë G.I. Shishkin. Approximation of solutions to singularly perturbed boundary value problems with corner boundary layers. Zh. Vychisl. Mat. Mat. Fiz., 27:1360í1374, èin Russianè. ë11ë G.I. Shishkin. Approximation of solutions to singularly perturbed boundary value problems with parabolic boundary layers. Zh. Vychisl. Mat. Mat. Fiz., 29:963í977, èin Russianè. ë12ë G.I. Shishkin. Grid approximation of singularly perturbed elliptic equation in domain with characteristic bounds. Sov. J. Numer. Anal. Math. Modelling., 5:327í343, ë13ë G.I. Shishkin. Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Ur. O. RAN, Ekaterinburg, èin Russianè. 13