Superconvergence analysis of Wilson element on anisotropic meshes

Applied Mathematics and Mechanics (English Edition), 2007, 28(1):119-125OEditorial Committee of Appl. Math. Mech., ISSN 0253 4827Superconvergence analysis of Wilson element on anisotropic meshesSHI Dong-yang (石东洋)'，LIANG Hui (梁慧)2(1. Department of Mathematics, Zhengzhou University, Zhengzhou 450052, P. R. China;2. Department of Mathematics, Harin Institute of Technology, Harbin 150001, P. R. China) .(Communicated by ZHONG Wan-xie)AbstractThe Wilson finite element method is considered to solve a class of twodimensional second order elliptic boundary value problems. By using of the particularstructure of the element and some new techniques, we obtain the superclose and globalsuperconvergence on anisotropic meshes. Numerical example is also given to confirm ourtheoretical analysis.Key words Anisotropic meshes, Wilson element, superclose, superconvergenceChinese Library Classifcation 0242.212000 Mathematics Subject Classification65N30, 65N15Digital Object Identifer(DOI) 10.1007/s 10483-007-0114-1IntroductionWilson element has very good numerical results in engineering computation and numerousstudies have been devoted to its convergence and superconvergence analysis (see Refs.[1-6)).Especially, Z. C. Shi, B. Jiang and W. M. Xuel2] have studied its superconvergence at vertices,midpoints of sides of the element on a class of so called uniform partition, i.e., all lengths ofedges of the element in x-direction are equal and so are all in y-direction. Q. Lin and N. N.Yanl4] have also studied its global superconvergence by using integral identities. However, as faras we know, all of the superconvergence results of Wilson element up to now are dependent ofthe restriction of classical regularity assumption hk/p≤c and the quasi-uniform assumptionh/h S c. Here, K is an element, hK and pK are the diameters of K and the biggest circlecontained in K respectively, c is a positive constant independent ofh = maxhk andh = min hk(see Ref.[1]) and the function under considered.It is well known that the solution of the elliptic boundary value problems may have anisotropicbehavior in parts of the domain. That is to say, the solution varies significantly only in certaindirections, such as singularly perturbed convection-diffusion-reaction problemns where bound-ary or the interior layers appear. Therefore, in view of both theoretical analysis and practicalapplication, the regularity assumption and quasi-uniform assumption are great drawbacks inthe finite element methods. In such cases, an obvious idea to reflect this anisotropy and over-come above fatal deficiency is to use anisotropic meshes in discretization. These are stretchedelements where the above aspect ratios can be very large or even unbounded. Although thisis converse to the conventional isotropy theory, the use of anisotropic discretization allows toachieve the same accuracy with less degrees of freedom and there have appeared some studiesfocusing on the study of anisotropic finite element methads rerontlv (os Rofe 6 -9]). However,the analysis of the superclose and superconvergence for中国煤化工on anisotropicmeshes has seldom been seen in the previous literatureY片CNMHG* Received Mar.1, 2005; Revised Sep.18, 2006Project supported by the National Natural Science Foundation of China (No. 10371113)Corresponding author SHI Dong yang, Professor, Doctor, E-mail: shi dy@zzu.edu.cn120SHI Dong-yang and LIANG HuiThe main aim of this paper is to study the superclose and global superconvergence of Wilson'selement on anisotropic meshes by employing some novel techniques and using some advantagesof Wilson element. Numerical example is also given to verify the validity of our theoreticalanalysis and the element performance. The results obtained in this paper can be regardedas a generalization to Ref.[4]. Moreover, our analysis will be helpful in developing posteriorestimates method and designing some adaptive algorithms of numerical solutions for the secondorder problems.Consider the following second order elliptic boundary value problem,f-Ou=f， in几,(1《u|r =0，)n T= 82,where s C R2 is rectangular domain, f∈L2(S).The weak form of Eq.(1) is to find u∈H2(R) = V, such thata(u,v)=f(v)，Vv∈ V, .(2)where a(u,)= so Vu . Vudxdy, f(v) = Sn fvdxdy.1 Superclose analysisFor the sake of simplicity, let∩C R2 be a bounded rectangular domain in x - -y plane with theboundary an paralleling to x-axis and y-axis respectively. Jh is a rectangular triangulation ofQ.VK∈Jh, assume the central point be (xK, yk) and four vertices bedi = (xk-hx,K,yk-hy,k),d2= (xK +hz,K,yk-hy,K),ds = (xK +hx,K, yK +hy,k) andds = (xk-hx,K, YK +hy,k), where2hx,K and 2hy,K are the lengths of two edges of K in x-direction and y-direction respectively.Lethx= max hx,K,hy= max hy.K, hmax = max(hx,hy). R = [-1, 1]x[-1, 1] be the referenceelement in ξ - η plane with verticesdi= (-1,-1),d2=(1,-1),d3=(1,1) andds=(-1,1).Let Vh be Wilson's finite element spacel2,4. Then the approximation problem of Eq.(2) is tofind Rhnu ∈V", such thatan(Rnu,vn)=f(vn)，Vvn∈ v,(3)where an(u,川) =三Jk Vu. Vvdxdy.Let Vh = {Rnu|Rnu ∈Q1(K) is determined by its function values at four vertices ofK,Rnulan= 0}, Rhu∈的be the solution satisfesan(Rru,Uh)= f(vn)，Vvn∈ 的.(4)It can be verifed thatan(Rhu- Rhu,vh)=an(u- Rru,vh)=0，VUn∈ 站。(5)In our forgoing analysis, the following anisotropic inverse inequality plays an essential role.Lemma1 V v∈V6h andV K∈In, there holdIl)l)o,x，≤ch; Illo,r，, Iyy)o, ≤chy Iyllok.(6)Proof We only prove the first inequality of Ref(6)中国煤化工lx1, K=u2dxdy =crh=4hzhyd&dMHCNMHhlei1R≤chglll = chjlyol1I.kh(hahy)-1 = chZ1vI1.K.Superconvergence of Wilson element121Lemma 2 Suppose u∈H3(2)∩H6(N) and inu is the bilinear intrpolation ofu, w =u- inu, then there holdwxUrdxdy = ch}|13lbl，I uyUydxdy = ch}ul3lvl，v∈ 沿(7)JsProof By Lemma 1 and the techniques provided in Ref.{4], we can get the desired results.Based on Lemma 1 and Lemma 2, we can deduce the following superclose property.Theorem 3 Supposeu∈ H3(M)∩H}(2) and Rnu are the soution of Eqs.(1) and (4)respectively, then|Rnu - inl|hn≤ch2.xlul3,(8)we临= Ekl. l.x.Proof By Lemma 2 and Eq.(5) we havea(Rnu- inu,) = a(u- inu,v) + a(Rru - u,v)≤chmaxlu|3|v|,taking U = Rhu - ihu in above inequality and using the coercive property of a(,) yield thedesired result.Lemma 4Let Rru be the solution of Eq.(3), under the assumption of Theorem 3,, wehaveinRnu= Rnu.(9)Proof Since 沿C V', VU∈的, noticing that |Rrul3,K = 0 and using of Theorem 3, wea(inRru- Rhu,v) = a(inRnu- Rnu,v)= 2 I (iRnu - Rau). Vvudxdy= O(h2)S |Rnyl},lIol|1,K= 0.(10)Taking U= inRhu- Rnu∈v的h in Eq.(10) yields the desired result.By Theorem 3 and Lemma 4, we can get the superclose property of Wilson's element.Theorem 5 Under the same assumptions of Theorem 3, we have|Rru - inulh≤ch2|ul3.(11)2 Superconvergence analysisIn order to get the global superconvergence, let J2h be a family of anisotropic meshes. Jhcan be obtained by dividing each element of J2h into four equal elements K;(i= 1,2,3,4). LetK∈J2h, k= U K; Then we can construct the post- processing interpolation operator啄on V的U H3(Q2) satisfying中国煤化工Ilh|k : Vlk UHr(k)(12)MYHCNMHGI2rv(ai) = v(a),i = 1,2...9. Here Q2(k) denotes the biquadratic polynomial space of xandyonK.Itiseasytoseethat122SHI Dong-yang and LIANG Huih(inw) = Ihw, Vw ∈V6 U H(2).(13)Lemma 6 For above强h，there hold|u- I吐ea≤chha -lul+t,t, Vu∈H(2)， 1≤γ≤2， l=0,1,(14)I(InI)≤col(h，V∈ Vh.(15)Proof Estimate (14) can be proved by the same method as in Ref.[3]. Since(I(I))1B,k= J<(TIqD)层dxdy =后mhkhk d&dd = helQAl，限≤chRhlWiH.k =chiR。Jk唱瓷。hn呢drdy = lk,(16)we have(17)similarly(18)By summing Eqs.(17) and (18) over all elements of Jh, we can get Eq.(15). The proof iscompleted.Theorem 7 Under the hypothesis of Theorem 3, there holds the following superconver-gence estirmate on anisotropic meshes|u - IninRrlh≤chaxlul}.Proof By Lemma 4 and Theorem 5, we have|IhinRru - ulh≤|u - h吼(nu)lh + mhinRmu - h(inu)hn≤|u - I,zlh + lI2n(inu - inRru)h≤c(hmaxlu4l3 + linu - inRrul|h)≤chMax1ul3.3 Numerical exampleIn order to verify our theoretical analysis and examine the performance of the Wilson elementon anisotropic meshes, we consider problem (1) with∩= [0,1]2C R2 andf=-Ou=-2(1-e-t)(1-x)-. 2-2-2+2-1+-3)1-)e-喜(1-x)x(1-y)②2 )u(1-),we can verify thatu=(1-x)y(1-y)(1-e~/&)+(1-y)x(1-x)(1-e-/e)中国煤化工is a solution of Eq.(1).Because u(xr, y) varies intensively near two edgesJYHCNMHGhenεissmallenough, we divide n with different meshes in different parts. We first determine the boundarylayer, which is denoted by a. Then we divide each edges of几into two parts according to a, andSuperconvergence of Wilson element123denote the boundary layer of the two edges ofS (x= 0 andy= 0) byS1 = (0,a) x (0,1) andR2 = (0, 1)x (0, a) respectively. In this way, s is composed of four parts: S1∩S2,几- h1 US2,821-∩1∩S2andS2-R1∩Q2.Let p be the ratio of the boundary layer's subdividing cost to the whole subdividing costalong one edge. For example, if we divide one edge into n small parts, then the boundary layertakes n . p equal small parts, and the remain part of this edge takes n.(1 - p) equal small parts.We can get different meshes when p varies. Here we takep= 1/2 and p= 3/5 to get two kindsof meshes (see Fig.1, Mesh 1 and Mesh 2).We consider two cases: ε = 0.05 and ε = 0.02. The graphs of solution u(x, y) are listedin Fig.2 and Fig.3 respectively. In following tables 1-8, a represents the average convergenceorder, m and n denote the number of subdividing element along x -direction and y-directionrespectively.In Table 9, let hr and hs denote the lengths of the longer edge and the shorter one of KMesh1(p-0.5)Mesh2(p-0.6)Fig.1 Two kinds of meshes0.3(0.30.2u0.n1.01..80.80.6y 0.40.4 xFig.2 u(x,y) (e = 0.05)Fig.3 u(x,y) (ε = 0.02)Table 1 Approximation results withε= 0.05,a= 0.18 andp= 1/2mxn|u - Rrullo .|u- Rrullh中国煤化工nRnulh40 x402.7514797 x 10-32.077063469 x 10- 1fHCNMHG0x10-380x 806.886549x 10-41.039749585 x 10-11.441417 x 10-3160x1601.722130x 10-45.20027540x 10-26.87865 x 10-53.140017x 10-41.998970.9989431.997971.97848124SHI Dong-yang and LIANG HuiTable 2 Approximation results withε = 0.05,a= 0.18 andp = 3/5mxn.|u - Rrullo .|u - Rnul|h|Rru - inl/h|u - IhinRrullr40x40 4.2956967x 10-32.595329278x 10-1 1.6669605x 10-3.4.3491402 x 10-380x 801.0749326x 10- 31.298887720 x 10- 14.177859x 10- 41.1667889 x 10- 3160x 1602.687965x 10- 46.49597409 x 10- 21.045139 x 10- 42.985274x 10~4a1.999150.999151.997731.9328Table 3 Approximation results withε = 0.05,a= 0.20 andp= 1/2mxn|u - Rnullo|u- RnullhllBu-inl|h |lu - IzgninRrllh40x40 2.5961414x 10- 32.011467162x 10-11.0287218x 10-35.6307887 x 10-36.499124 x 10-41.007135721 x 10- 12.577927 x 10-41.4254743 x 10-31.625331 x 10-45.03743762 x 10-26.44870 x 10-53.576058 x 10-41.998780.9987431.997851.98846Table 4 Approximation results with ε= 0.05,a= 0.20 andp= 3/5mx n|u - Rrullo|u- Rrul/h|Rru- int//hIu - IhnRrulh40x404.0520037 x 10-32.512377089 x 10-11.5479066x 10-3 4.3986615x 10-380x801.0140848x 10-3 1.257552398 x 10-13.879344x 10-41.1337281 x 10-32.535892 x 10-46.28947300x 10-29.70451 x 10-52.863042 x 10-41.999040.9990211.997761.9708Table 5 Approximation results with ε= 0.02,a= 0.05 andp= 1/2|u- Rrullo|u - Rrullh|Rru- inrlh |u - IhinRulh1.37779677 x 10-2 9.131560630x 10-1 2.8023323 x 10-3 3.99815970x 10-23.4493909 x 10-34.571854644 x 10-17.085860 x 10-41.66707959x 10-2160x 160 8.627183x 10-4 2.286826074x 10-1 1.777241 x 10-45.1238532 x 10-31.998670.9987581.989471.49873Table 6 Approximation results withε= 0.02,a= 0.07 andp= 1/2u - Ru/h|IRyu - inullh肌-骚hi剧ruh40x 401.31839998x 10-2 8.931047108 x 10-12.4506205x 10-3 1.55100272 x 10-23.3004158x 10-34.471448302 x 10-16.161844 x 10-6.0387176x 10-8160x 160 8.253884 x 10-42.236483305 x 10-11.542925 x 10-41.8218848x 10-31.998790.9987991.994711.55655Table 7 Approximation results withε= 0.02,a= 0.18 and p = 3/5|u - Rru(h|Rnu - inulh|u - II场iRrllh40 x401.56001908 x 10-29.598546147 x 10-1中国煤化工182x 10-23.9182545x 10- 34.821773487 x 10- 1TYHCNM H G99x 10-3160 x 1609.807140x 10-42.413729924 x 10-11.673196 x 10-1.7800004 x 10-31.99580.9957781.993821.97639Superconvergence of Wilson element125Table 8 Approximation results with ε= 0.02,a= 0.20 and p= 3/5mx n儿u - Rrullo|u- Rru|/h|Rru - inulh|u - I3nin Rnullh40x 401.47167111 x 10-29.284784397 x 10-12.5023655x 10-3 3.36911149 x 10-280x 803.7000743x 10-3 4.668842181 x 10-16.305381 x 10-48.6972803 x 10- 3160 x 1609.263422 x 10-42.337771517 x 10-1.579589x 10-42.1923313 x 10-3a1.994890.994871.992841.97102It can be seen from above tables that theTable 9 Values of max(hr/hs)numerical results rely on the choices of a andp=1/2p=3/5p which ensure that the subdivisions are consis-tent with the physical characters of real solutiona= 0.18455555656.833333u(x, y). Meanwhile, on anisotropic meshes, whena= 0.20.4.0000006.00000h→0, |u-IIZnihRrul|h, |u- Rrul|h and |Rnu-a= 0.0519.000002400000inu|h converge at rates of O(h2), O(h) and 0(h2)a= 0.0713.28570417.14286respectively, which coincide with our theoreticalanalysis.References[] Ciarlet P G. The finite element method for eliptice problem[M]. Amsterdam: North-Holland, 1978.[2} Shi Z C, Jiang B, Xue W M. A new superconvergence property of Wilson nonconforming finiteelement[J]. Numer Math, 1997, 78:259- 268.[3] Luo Ping, Lin Qun. High accuracy analysis of the Wilson element[J]. J of Comput Math, 1999,17(2):113-124.[4] Lin Qun, Yan Ningning. The construction and analysis of high accurate finite element methods[M].Shijiazhuang: Hebei University Press, 1996 (in Chinese).] Shi Dongyang, Chen Shaochun. A kind of improved Wilson arbitrary quadrilateral elements[J]. JNumer Math, J Chinese University, 1994, 16(2):161-167 (in Chinese).[6] Chen Shaochun, Zhao Yongcheng, Shi Dongyang. Anisotropic interpolation with application tononconforming elements([J]. Appl Numer Math, 2004, 49:135 152.[7]Zienissek A, Vanmaele M. The interpolation theorem for narrow quadrilateral isotropic metricfinite elements[J]. Numer Math, 1995, 72:123 -141.] Apel Th, Dobrowolski M. Anisotropic interpolation with applications to the finite elementmethod[J]. Computing, 1992, 47:277- 293.[9] Apel Th. Anisotropic finite elements: local estimate and applications[M]. Stuttgart, Leipzig:B.G.Teubner, 1999.中国煤化工MYHCNMHG