American Journal of Computational Mathematics, 2012, 2, 249257 http://dx.doi.org/10.4236/ajcm.2012.24034 Published Online December 2012 (http://www.SciRP.org/journal/ajcm) Application for Superconvergence of Finite Element Approximations for the Elliptic Problem by Global and Local L2Projection Methods Rabeea H. Jari, Lin Mu Department of Applied Science, UALR, Little Rock, USA Email: rhjari@ualr.edu, lxmu@ualr.edu Received February 12, 2012; revised May 21, 2012; accepted July 12, 2012 ABSTRACT Numerical experiments are given to verify the theoretical results for superconvergence of the elliptic problem by global and local L2Projection methods. Keywords: Finite Element Methods; Superconvergence; L2Projection; Elliptic Problem 1. Introduction The elliptic problem seeks u in a certain functional space such that in uf (1) in ug (2) where denote the Laplacian operator. Let h be a finite element partition of the domain T with characteristic mesh size h. Let be any finite element space for u associated with the partition . 1 hg VH h T The L2Projection technique was introduced by Wang [13]. It projects the approximate solution to another finite element dimensional space associated with a coarse mesh. Now, we start with defining a coarse mesh T where h satisfying: h (3) with . Define finite element space . Let 0,1 2s VH Q to be the L2Projector onto the finite element space V [1,4,5]. The Projector Q can be considered as a linear operator (projection) from onto the finite element space 2 L V [6,7]. 2. Superconvergence by Global L2Projection The following theorems can be found in [1]. Theorem 2.1: Assume that 11 k and the finite element space . If the exact solution 2s VH 1 11krg H H uH , then there exists a constant C such that 1, hh rh uQu huQu ChuChu u where 1min0,2 s and is the finite element approximation of (1) and (2). h u Theorem 2.2: Suppose that 11 k u. Let the sur face fitting spaces and h be the finite element approximation of (1) and (2). Then, the post processing of is estimated by 2s VH h u 1 1min0,2 ks rs . 3. Numerical Experiments for Global L2Projection In this section, we present several numerical experiments to verify the theoretical analysis in [1]. The triangulation is constructed by: 1) dividing the domain into an h T n 3 n 3 rectangular mesh; 2) connecting the diagonal line with the positive slope. Denote 3 1 hn as the mesh size. The finite element space is defined by 1 1 ;; , on hg h K VvH vPKKTvg. We define V as follows: 2 2 :; K VvLvPKKT . Example 3.1: Let the domain 0,1 0,1 and the exact solution is assumed as C opyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 250 11ux xyy . Table 1 shows that after the postprocessing method, all the errors are reduced. The exact solution in L2norm of h uQu has the similar convergence rate as h uu. There is no improvement for the u in L2norm. However, the error in H1norm have higher convergence rate, which is shown as 1.3 Oh for h uQu . The order of convergence rate is better than 0.3 Oh , h uu see Figures 1(a) and (b). Figures 2(a) and (b) give results for the finite element approximation of (1)(2) before and after postprocessing. Example 3.2: Let the domain 0,1 0,1 and the exact solution is assumed as sin πcos π.uxy Table 1. Errors on uniform triangular meshes Th and Tτ. h 1 h uu h uu 1 h uQu h uQu 2−3 0.6632e−2 0.1287e−3 0.1427e−2 0.1227e−3 3−3 0.2799e−2 0.2295e−4 0.4332e−3 0.2185e−4 4−3 0.1433e−2 0.6017e−5 0.1763e−3 0.5730e−5 5−3 0.8294e−3 0.2015e−5 0.8504e−4 0.1919e−5 6−3 0.5223e−3 0.7992e−6 0.4596e−4 0.7610e−6 Oh 0.9998 1.9993 1.3504 1.9996 (a) (b) Figure 1. (a) Convergence rate of L2norm error; (b) Convergence rate of H1norm error. (a) (b) Figure 2. (a) Surface plot of approximation solution uh; (b) Surface plot of approximation solution Qτuh. Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 251 From the results shown in Table 2, it is clear that the exact solution u in H1norm has the superconvergence, but there is no improvement in the L2norm, see Figures 3(a) and (b). The finite element solution given in Fig ures 4(a) and (b). This agrees well with the theory. Example 3.3: Let the domain 0,1 0,1 and the exact solution is assumed as cos π . 2 y u Table 3 gives the errors profile for Example 3. Notice that, the gradient estimate is of order , that is 1.3 Oh much better than the optimal order . Although, there is no improvement in the L2norm, see Figure 5. Oh Figure 6 shows that the approximation solutions and . h u h Also, our numerical results and theoretical conclusions in Theorems (2.1) and (2.2) show highly consistent. Qu Table 2. Errors on uniform triangular meshes Th and Tτ. h 1 h uu h uu 1 h uQu h uQu 2−3 0.9629e−1 0.1598e−2 0.2242e−1 0.1498e−2 3−3 0.4063e−1 0.2850e−3 0.6872e−2 0.2669e−3 4−3 0.2080e−1 0.7475e−4 0.2810e−2 0.6998e−4 5−3 0.1204e−1 0.2503e−4 0.1359e−2 0.2343e−4 6−3 0.7582e−2 0.9929e−5 0.7363e−3 0.9294e−5 Oh 0.9998 1.9991 1.3427 1.9995 (a) (b) Figure 3. (a) Convergence rate of error L2norm error; (b) Convergence rate of H1norm error. (a) (b) Figure 4. (a) Surface plot of solution uh; (b) Surface plot of approximation solution Qτuh. Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 252 Table 3. Errors on uniform triangular meshes Th and Tτ. h 1 h uu h uu 1 h uQu h uQu 2−3 0.9135e−1 0.1770e−2 0.2150e−1 0.1689e−2 3−3 0.3855e−1 0.3157e−3 0.6579e−2 0.3010e−3 4−3 0.1973e−1 0.8278e−4 0.2692e−2 0.7893e−4 5−3 0.1142e−1 0.2772e−4 0.1303e−2 0.2643e−4 6−3 0.7193e−2 0.1099e−4 0.7062e−3 0.1048e−4 Oh 0.9999 1.9993 1.3424 1.9994 (a) (b) Figure 5. (a) Convergence rate of L2norm error; (b) Convergence rate of H1norm error. (a) (b) Figure 6. (a) Surface plot of approximation solution uh; (b) Surface plot of approximation solution Qτuh. 4. Superconvergence by Local L2Projection Notice that, the exact solution u may be not smooth globally on in practical computation, although the solution might be smooth enough locally for a good su per convergence. To this end, let be a subdomain of where the 0 exact solution u is sufficiently smooth. Let be an 1 other subdomain of such that 01 . Define fi nite element space The L2projection 2 1 s VH Q from 2 L onto the finite element space V is said to be local L2projection. The following theorem can be found in [1]. Theorem 4.1: Assume that 11 k and the finite element space 2 0 s VH . If the exact solution 11 0 kr H H 1 , g uH then there exists a Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 253 constant C such that 00 00 (1) , hh rh uQu huQu ChuChu u where is the finite element approximation of (1)(2). h Theorem 4.2: Suppose that 1 u 1 k. Let the sur face fitting spaces 0 u 2s VH and h be the finite element approximation of (1)(2). Then, the postproc essing of is estimated by h u 1. 1min0,2 ks rs 5. Numerical Experiments for Local L2Projection In this section, we present several numerical experiments to verify the theoretical analysis in [1]. The triangulation is constructed by: 1) dividing the domain into an h T 3 nn3 rectangular mesh; 2) connecting the diagonal line with the positive slope. Denote 3 1 hn as the mesh size. The finite element space is defined by 1 1 ;;, on . hg h K VvH vPKKTvg We define V as follows: 2 2 :; K VvL vPKKT . Example 5.1: Let the domain 0,1 0,1 and 00, 0.50, 0.5 . The exact solution is assumed as 1. 2 u y It is clear that the exact solution u is singular and f blows down at the boundary of 0,1 0,1 , see Figure 7, however, h and h Qu are sufficiently smooth on u 0,1 0,1 , see Figure 8. Table 4 shows that after the postprocessing method, all the errors are reduced. The exact solution in L2norm of h uQu has the similar convergence rate as h uu which is shown as . There is no im provement for the u in L2norm. However, the error in H1norm have higher convergence rate, which is shown 2 Oh as 1.3 Oh for h uQu . The order of conver gence rate is 0.3 Oh better than hh uu, see Figure 9. Example 5.2: Let the domain 0,1 0,1 and 00.5,1 0.5,1 . The exact solution is assumed as 22 uxy Obviously, the exact solution has singularity on the origin at the domain 0,1 0,1 , see Figure 10(a). On the same domain the function f blows down at the boundary, see Figure 10(b). The approximation solu tions u and have been plot in the proper subdo main h Qu 0 From the results shown in Table 5, it is clear that the exact u in H1norm has the superconvergence, but there is no improvement in the L2norm, see Figure 12. This agrees well with the theory. 0.5,1 0.5,1 , see Figure 11. Example 6: Let the domain 0,1 0,1 and 00.5,1 0.5,1 . The exact solution is assumed as 22 . y u y From Figures 13(a) and (b), respectively observe that the exact solution has strongly singularity on the origin of the domain 0,1 0,1 h uh Qu and the function f blows up at the boundary, Figure 14 show how the approxima tion solution and look like at the proper sub domain 0 0.5,1 0.5,1. (a) (b) Figure 7. (a) The exact solution u blows up; (b) f blows down at the boundary. Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 254 (a) (b) Figure 8. (a) Surface plot of approximation solution uh; (b) Surface plot of approximation solution Qτuh. Table 4. Errors on uniform triangular meshes Th and Tτ. h 1 h uu h uu 1 h uQu h uQu 2−3 0.3221e−1 0.1497e−2 0.1026e−1 0.1363e−2 3−3 0.1291e−1 0.2384e−3 0.2566e−2 0.2169e−3 4−3 0.8072e−2 0.9306e−4 0.1429e−2 0.8466e−4 5−3 0.5871e−2 0.4921e−4 0.9977e−3 0.4476e−4 6−3 0.4613e−2 0.3037e−4 0.7691e−3 0.2763e−4 Oh 0.9998 2.0030 1.3360 2.0035 (a) (b) Figure 9. (a) Convergence rate of L2norm error; (b) Convergence rate of H1norm error. Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 255 (a) (b) Figure 10. (a) Surface plot of exact solution u; (b) f blows down at the boundary. (a) (b) Figure 11. (a) Surface plot of approximation solution uh; (b) Surface plot of approximation solution Qτuh. Table 5. Errors on uniform triangular meshes Th and Tτ. h 1 h uu h uu 1 h uQu h uQu 2−3 0.1352e−1 0.1400e−2 0.6141e−2 0.1287e−2 3−3 0.6835e−2 0.3596e−3 0.2110e−2 0.3314e−3 4−3 0.4566e−2 0.1607e−3 0.1215e−2 0.1481e−3 5−3 0.3427e−2 0.9058e−4 0.8529e−3 0.8352e−4 6−3 0.2743e−2 0.5802e−4 0.6590e−3 0.5350e−4 Oh 0.9923 1.9806 1.3581 1.9792 Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU 256 (a) (b) Figure 12. (a) Convergence rate of L2norm error; (b) Convergence rate of H1norm error. (a) (b) Figure 13. (a) Surface plot of exact solution u; (b) f blows up at the boundary . (a) (b) Figure 14. (a) Surface plot of approximation solution uh; (b) Surface plot of approximation solution Qτuh. Copyright © 2012 SciRes. AJCM
R. H. JARI, L. MU Copyright © 2012 SciRes. AJCM 257 Table 6. Errors on uniform triangular meshes Th and Tτ. h 1 h uu h uu 1 h uQu h uQu 2−3 0.1186e−1 0.4006e−3 0.4708e−2 0.2779e−3 3−3 0.5979e−2 0.1009e−3 0.1621e−2 0.6959e−4 4−3 0.3992e−2 0.4490e−4 0.9518e−3 0.3094e−4 5−3 0.2996e−2 0.2527e−4 0.6760e−3 0.1740e−4 6−3 0.2397e−2 0.1617e−4 0.5261e−3 0.1113e−4 Oh 0.9943 1.9949 1.3304 1.9989 (a) (b) Figure 15. (a) Convergence rate of L2norm error; (b) Convergence rate of H1norm error. Table 6 gives the errors profile for Example 6. Notice that, the gradient estimate is of order that is 1.3 Oh much better than the optimal order . Although, there is no improvement in the L2norm, see Figure 15. Also, the numerical results and theoretical conclusions show highly consistent. Oh REFERENCES [1] J. Wang, “A Superconvergence Analysis for Finite Ele ment Solutions by the LeastSquares Surface Fitting on Irregular Meshes for Smooth Problems,” Journal of Mathematical Study, Vol. 33, No. 3, 2000, pp. 229243. [2] R. E. Ewing, R. Lazarov and J. Wang, “Superconver gence of the Velocity along the Gauss Lines in Mixed Fi nite Element Methods,” SIAM Journal on Numerical Analysis, Vol. 28, No. 4, 1991, pp. 10151029. doi:10.1137/0728054 [3] M. Zlamal, “Superconvergence and Reduced Integration in the Finite Element Method,” Mathematics Computa tion, Vol. 32, No. 143, 1977, pp. 663685. doi:10.2307/2006479 [4] L. B. Wahlbin, “Superconvergence in Galerkin Finite Element Methods,” Lecture Notes in Mathematics, Springer, Berlin, 1995. [5] A. H. Schatz, I. H. Sloan and L. B. Wahlbin, “Supercon vergence in Finite Element Methods and Meshes that Are Symmetric with Respect to a Point,” SIAM Journal on Numerical Analysis, Vol. 33, No. 2, 1996, pp. 505521. doi:10.1137/0733027 [6] M. Krizaek and P. Neittaanmaki, “Superconvergence Phenomenon in the Finite Element Method Arising from Avaraging Gradients,” Numerische Mathematik, Vol. 45, No. 1, 1984, pp. 105116. [7] J. Douglas and T. Dupont, “Superconvergence for Galer kin Methods for the TwoPoint Boundary Problem via Local Projections,” Numerical Mathematics, Vol. 21, No. 3, 1973, pp. 270278. doi:10.1007/BF01436631
