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January  2019, 24(1): 19-54. doi: 10.3934/dcdsb.2018104

Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems

Mahboub Baccouch

Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182, USA

Received  March 2017 Revised  January 2018 Published  March 2018

In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the first-and second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p+3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

Keywords: Local discontinuous Galerkin method KdV equations, stability a priori error estimates superconvergence Gauss-Radau projections.
Mathematics Subject Classification: Primary: 65M15, 65M60, 65M50, 65N30, 65N50.
Citation: Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. Google Scholar

[2]

S. Adjerid and M. Baccouch, A superconvergent local discontinuous Galerkin method for elliptic problems, Journal of Scientific Computing, 52 (2012), 113-152. doi: 10.1007/s10915-011-9537-8. Google Scholar

[3]

S. Adjerid and D. Issaev, Superconvergence of the local discontinuous Galerkin method applied to diffusion problems, in: K. Bathe (ed. ), Proceedings of the Third MIT Conference on Computational Fluid and Solid Mechanics, vol. 3, Elsevier, 2005.Google Scholar

[4]

S. Adjerid and A. Klauser, Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems, Journal of Scientific Computing, 22 (2005), 5-24. doi: 10.1007/s10915-004-4133-9. Google Scholar

[5]

M. Baccouch, A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209/212 (2012), 129-143. doi: 10.1016/j.cma.2011.10.012. Google Scholar

[6]

M. Baccouch, Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems, Applied Mathematics and Computation, 226 (2014), 455-483. doi: 10.1016/j.amc.2013.10.026. Google Scholar

[7]

M. Baccouch, Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension, Computers & Mathematics with Applications, 67 (2014), 1130-1153. doi: 10.1016/j.camwa.2013.12.014. Google Scholar

[8]

M. Baccouch, A superconvergent local discontinuous Galerkin method for the second-order wave equation on cartesian grids, Computers and Mathematics with Applications, 68 (2014), 1250-1278. doi: 10.1016/j.camwa.2014.08.023. Google Scholar

[9]

M. Baccouch, Asymptotically exact local discontinuous Galerkin error estimates for the linearized Korteweg-de Vries equation in one space dimension, International Journal of Numerical Analysis and Modeling, 12 (2015), 162-195. Google Scholar

[10]

M. Baccouch, Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension, Journal of Computational Mathematics, 34 (2016), 511-531. doi: 10.4208/jcm.1603-m2015-0317. Google Scholar

[11]

T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. Google Scholar

[12]

J. L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Mathematics of Computation, 82 (2013), 1401-1432. doi: 10.1090/S0025-5718-2013-02661-0. Google Scholar

[13]

W. Cao and Z. Zhang, Superconvergence of local discontinuous Galerkin methods for one-dimensional linear parabolic equations, Mathematics of Computation, 85 (2016), 63-84. Google Scholar

[14]

P. Castillo, A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 4675-4685. doi: 10.1016/S0045-7825(03)00445-6. Google Scholar

[15]

P. Castillo, A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, 56 (2006), 1307-1313. doi: 10.1016/j.apnum.2006.03.016. Google Scholar

[16]

P. Castillo, B. Cockburn, D. Schötzau and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Mathematic of Computation, 71 (2002), 455-478. Google Scholar

[17]

F. Celiker and B. Cockburn, Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Mathematics of Computation, 76 (2007), 67-96. doi: 10.1090/S0025-5718-06-01895-3. Google Scholar

[18]

Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), 4044-4072. doi: 10.1137/090747701. Google Scholar

[19]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978. Google Scholar

[20]

B. Cockburn, G. Kanschat and D. Schötzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Mathematics of Compuatation, 74 (2005), 1067-1095. Google Scholar

[21]

B. Cockburn, G. Kanschat and D. Schötzau, The local discontinuous Galerkin method for linearized incompressible fluid flow: A review, Computers & Fluids, 34 (2005), 491-506. doi: 10.1016/j.compfluid.2003.08.005. Google Scholar

[22]

B. Cockburn, G. E. Karniadakis and C. W. Shu, Discontinuous Galerkin Methods Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000.Google Scholar

[23]

B. Cockburn and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws Ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435. Google Scholar

[24]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. Google Scholar

[25]

T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2010. Google Scholar

[26]

K. D. Devine and J. E. Flaherty, Parallel adaptive hp-refinement techniques for conservation laws, Computer Methods in Applied Mechanics and Engineering, 20 (1996), 367-386. doi: 10.1016/0168-9274(95)00103-4. Google Scholar

[27]

J. E. Flaherty, R. Loy, M. S. Shephard, B. K. Szymanski, J. D. Teresco and L. H. Ziantz, Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws, Journal of Parallel and Distributed Computing, 47 (1997), 139-152. doi: 10.1006/jpdc.1997.1412. Google Scholar

[28]

C. Hufford and Y. Xing, Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation, Journal of Computational and Applied Mathematics, 255 (2014), 441-455. doi: 10.1016/j.cam.2013.06.004. Google Scholar

[29]

V. Kucera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA Journal of Numerical Analysis, 34 (2014), 820-861. doi: 10.1093/imanum/drt007. Google Scholar

[30]

X. Meng, C.-W. Shu, Q. Zhang and B. Wu, Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM Journal on Numerical Analysis, 50 (2012), 2336-2356. doi: 10.1137/110857635. Google Scholar

[31]

S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM Journal on Numerical Analysis, 21 (1984), 217-235. doi: 10.1137/0721016. Google Scholar

[32]

T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis, 28 (1991), 133-140. doi: 10.1137/0728006. Google Scholar

[33]

B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008.Google Scholar

[34]

A. Samii, N. Panda and C. Craig Michoskiand Dawson, A hybridized discontinuous galerkin method for the nonlinear Korteweg-de Vries equation, Journal of Scientific Computing, 68 (2016), 191-212. doi: 10.1007/s10915-015-0133-1. Google Scholar

[35]

D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method, SIAM Journal on Numerical Analysis, 38 (2000), 837-875. doi: 10.1137/S0036142999352394. Google Scholar

[36]

H. Wang, C.-W. Shu and Q. Zhang, Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems, Applied Mathematics and Computation, 272 (2016), 237-258, recent Advances in Numerical Methods for Hyperbolic Partial Differential Equations. doi: 10.1016/j.amc.2015.02.067. Google Scholar

[37]

Y. Xing, C.-S. Chou and C.-W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problems and Imaging, 7 (2013), 967-986. doi: 10.3934/ipi.2013.7.967. Google Scholar

[38]

Y. Xu and C.-W. Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3805-3822. doi: 10.1016/j.cma.2006.10.043. Google Scholar

[39]

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Communications in Computational Physics, 7 (2010), 1-46. Google Scholar

[40]

Y. Xu and C.-W. Shu, Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations, SIAM Journal on Numerical Analysis, 50 (2012), 79-104. doi: 10.1137/11082258X. Google Scholar

[41]

J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, 40 (2002), 769-791. doi: 10.1137/S0036142901390378. Google Scholar

[42]

Y. Yang and C.-W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM Journal on Numerical Analysis, 50 (2012), 3110-3133. doi: 10.1137/110857647. Google Scholar

[43]

Y. Yang and C.-W. Shu, Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations, Journal of Computational Mathematics, 33 (2015), 323-340. doi: 10.4208/jcm.1502-m2014-0001. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. Google Scholar

[2]

S. Adjerid and M. Baccouch, A superconvergent local discontinuous Galerkin method for elliptic problems, Journal of Scientific Computing, 52 (2012), 113-152. doi: 10.1007/s10915-011-9537-8. Google Scholar

[3]

S. Adjerid and D. Issaev, Superconvergence of the local discontinuous Galerkin method applied to diffusion problems, in: K. Bathe (ed. ), Proceedings of the Third MIT Conference on Computational Fluid and Solid Mechanics, vol. 3, Elsevier, 2005.Google Scholar

[4]

S. Adjerid and A. Klauser, Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems, Journal of Scientific Computing, 22 (2005), 5-24. doi: 10.1007/s10915-004-4133-9. Google Scholar

[5]

M. Baccouch, A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209/212 (2012), 129-143. doi: 10.1016/j.cma.2011.10.012. Google Scholar

[6]

M. Baccouch, Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems, Applied Mathematics and Computation, 226 (2014), 455-483. doi: 10.1016/j.amc.2013.10.026. Google Scholar

[7]

M. Baccouch, Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension, Computers & Mathematics with Applications, 67 (2014), 1130-1153. doi: 10.1016/j.camwa.2013.12.014. Google Scholar

[8]

M. Baccouch, A superconvergent local discontinuous Galerkin method for the second-order wave equation on cartesian grids, Computers and Mathematics with Applications, 68 (2014), 1250-1278. doi: 10.1016/j.camwa.2014.08.023. Google Scholar

[9]

M. Baccouch, Asymptotically exact local discontinuous Galerkin error estimates for the linearized Korteweg-de Vries equation in one space dimension, International Journal of Numerical Analysis and Modeling, 12 (2015), 162-195. Google Scholar

[10]

M. Baccouch, Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension, Journal of Computational Mathematics, 34 (2016), 511-531. doi: 10.4208/jcm.1603-m2015-0317. Google Scholar

[11]

T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. Google Scholar

[12]

J. L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Mathematics of Computation, 82 (2013), 1401-1432. doi: 10.1090/S0025-5718-2013-02661-0. Google Scholar

[13]

W. Cao and Z. Zhang, Superconvergence of local discontinuous Galerkin methods for one-dimensional linear parabolic equations, Mathematics of Computation, 85 (2016), 63-84. Google Scholar

[14]

P. Castillo, A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 4675-4685. doi: 10.1016/S0045-7825(03)00445-6. Google Scholar

[15]

P. Castillo, A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, 56 (2006), 1307-1313. doi: 10.1016/j.apnum.2006.03.016. Google Scholar

[16]

P. Castillo, B. Cockburn, D. Schötzau and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Mathematic of Computation, 71 (2002), 455-478. Google Scholar

[17]

F. Celiker and B. Cockburn, Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Mathematics of Computation, 76 (2007), 67-96. doi: 10.1090/S0025-5718-06-01895-3. Google Scholar

[18]

Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), 4044-4072. doi: 10.1137/090747701. Google Scholar

[19]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978. Google Scholar

[20]

B. Cockburn, G. Kanschat and D. Schötzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Mathematics of Compuatation, 74 (2005), 1067-1095. Google Scholar

[21]

B. Cockburn, G. Kanschat and D. Schötzau, The local discontinuous Galerkin method for linearized incompressible fluid flow: A review, Computers & Fluids, 34 (2005), 491-506. doi: 10.1016/j.compfluid.2003.08.005. Google Scholar

[22]

B. Cockburn, G. E. Karniadakis and C. W. Shu, Discontinuous Galerkin Methods Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000.Google Scholar

[23]

B. Cockburn and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws Ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435. Google Scholar

[24]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. Google Scholar

[25]

T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2010. Google Scholar

[26]

K. D. Devine and J. E. Flaherty, Parallel adaptive hp-refinement techniques for conservation laws, Computer Methods in Applied Mechanics and Engineering, 20 (1996), 367-386. doi: 10.1016/0168-9274(95)00103-4. Google Scholar

[27]

J. E. Flaherty, R. Loy, M. S. Shephard, B. K. Szymanski, J. D. Teresco and L. H. Ziantz, Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws, Journal of Parallel and Distributed Computing, 47 (1997), 139-152. doi: 10.1006/jpdc.1997.1412. Google Scholar

[28]

C. Hufford and Y. Xing, Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation, Journal of Computational and Applied Mathematics, 255 (2014), 441-455. doi: 10.1016/j.cam.2013.06.004. Google Scholar

[29]

V. Kucera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA Journal of Numerical Analysis, 34 (2014), 820-861. doi: 10.1093/imanum/drt007. Google Scholar

[30]

X. Meng, C.-W. Shu, Q. Zhang and B. Wu, Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM Journal on Numerical Analysis, 50 (2012), 2336-2356. doi: 10.1137/110857635. Google Scholar

[31]

S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM Journal on Numerical Analysis, 21 (1984), 217-235. doi: 10.1137/0721016. Google Scholar

[32]

T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis, 28 (1991), 133-140. doi: 10.1137/0728006. Google Scholar

[33]

B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008.Google Scholar

[34]

A. Samii, N. Panda and C. Craig Michoskiand Dawson, A hybridized discontinuous galerkin method for the nonlinear Korteweg-de Vries equation, Journal of Scientific Computing, 68 (2016), 191-212. doi: 10.1007/s10915-015-0133-1. Google Scholar

[35]

D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method, SIAM Journal on Numerical Analysis, 38 (2000), 837-875. doi: 10.1137/S0036142999352394. Google Scholar

[36]

H. Wang, C.-W. Shu and Q. Zhang, Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems, Applied Mathematics and Computation, 272 (2016), 237-258, recent Advances in Numerical Methods for Hyperbolic Partial Differential Equations. doi: 10.1016/j.amc.2015.02.067. Google Scholar

[37]

Y. Xing, C.-S. Chou and C.-W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problems and Imaging, 7 (2013), 967-986. doi: 10.3934/ipi.2013.7.967. Google Scholar

[38]

Y. Xu and C.-W. Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3805-3822. doi: 10.1016/j.cma.2006.10.043. Google Scholar

[39]

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Communications in Computational Physics, 7 (2010), 1-46. Google Scholar

[40]

Y. Xu and C.-W. Shu, Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations, SIAM Journal on Numerical Analysis, 50 (2012), 79-104. doi: 10.1137/11082258X. Google Scholar

[41]

J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, 40 (2002), 769-791. doi: 10.1137/S0036142901390378. Google Scholar

[42]

Y. Yang and C.-W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM Journal on Numerical Analysis, 50 (2012), 3110-3133. doi: 10.1137/110857647. Google Scholar

[43]

Y. Yang and C.-W. Shu, Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations, Journal of Computational Mathematics, 33 (2015), 323-340. doi: 10.4208/jcm.1502-m2014-0001. Google Scholar

Figure 1.  Space-time graphs of the exact solution u (left) and the LDG solution uh (right) for Example 105 using N = 80 and p = 3.
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Figure 2.  Space-time graphs of the exact solution u (left) and the LDG solution uh (right) for Example 4 using N = 80 and p = 3.
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Table 1.  The $L^2$ errors $||e_u||$ and their orders of convergence for Example 1 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||e_u||$ order $||e_u||$ order $||e_u||$ order $||e_u||$ order
10 4.5846e-02 1.8733e-03 3.0396e-05 9.4387e-07
20 1.0865e-02 2.0771 1.1247e-04 4.0580 1.8971e-06 4.0020 2.9595e-08 4.9952
30 4.7435e-03 2.0440 3.0068e-05 3.2536 3.7529e-07 3.9964 3.9006e-09 4.9979
40 2.6395e-03 2.0376 1.2436e-05 3.0689 1.1883e-07 3.9975 9.2598e-10 4.9987
50 1.6794e-03 2.0263 6.3325e-06 3.0245 4.8695e-08 3.9980 3.0349e-10 4.9990
60 1.1623e-03 2.0186 3.6576e-06 3.0105 2.3490e-08 3.9984 1.2198e-10 4.9994
70 8.5246e-04 2.0112 2.3015e-06 3.0052 1.2682e-08 3.9986 5.6441e-11 4.9994
80 6.5263e-04 2.0004 1.5413e-06 3.0026 7.4350e-09 3.9989 2.8952e-11 4.9992
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p=1 p=2 p=3 p=4
N $||e_u||$ order $||e_u||$ order $||e_u||$ order $||e_u||$ order
10 4.5846e-02 1.8733e-03 3.0396e-05 9.4387e-07
20 1.0865e-02 2.0771 1.1247e-04 4.0580 1.8971e-06 4.0020 2.9595e-08 4.9952
30 4.7435e-03 2.0440 3.0068e-05 3.2536 3.7529e-07 3.9964 3.9006e-09 4.9979
40 2.6395e-03 2.0376 1.2436e-05 3.0689 1.1883e-07 3.9975 9.2598e-10 4.9987
50 1.6794e-03 2.0263 6.3325e-06 3.0245 4.8695e-08 3.9980 3.0349e-10 4.9990
60 1.1623e-03 2.0186 3.6576e-06 3.0105 2.3490e-08 3.9984 1.2198e-10 4.9994
70 8.5246e-04 2.0112 2.3015e-06 3.0052 1.2682e-08 3.9986 5.6441e-11 4.9994
80 6.5263e-04 2.0004 1.5413e-06 3.0026 7.4350e-09 3.9989 2.8952e-11 4.9992
Table 2.  The $L^2$ errors $||e_q||$ and their orders of convergence for Example 1 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||e_q||$ order $||e_q||$ order $||e_q||$ order $||e_q||$ order
10 4.0632e-02 2.1357e-03 8.1067e-05 2.5683e-06
20 1.0377e-02 1.9692 2.6569e-04 3.0069 5.1201e-06 3.9849 8.0223e-08 5.0007
30 4.6494e-03 1.9801 7.9028e-05 2.9905 1.0154e-06 3.9902 1.0585e-08 4.9952
40 2.6263e-03 1.9854 3.3377e-05 2.9961 3.2190e-07 3.9933 2.5136e-09 4.9976
50 1.6852e-03 1.9884 1.7106e-05 2.9956 1.3200e-07 3.9949 8.2408e-10 4.9977
60 1.1723e-03 1.9905 9.9056e-06 2.9965 6.3705e-08 3.9959 3.3129e-10 4.9982
70 8.6237e-04 1.9918 6.2400e-06 2.9978 3.4405e-08 3.9965 1.5331e-10 4.9986
80 6.6088e-04 1.9929 4.1817e-06 2.9975 2.0176e-08 3.9969 7.8645e-11 4.9990
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p=1 p=2 p=3 p=4
N $||e_q||$ order $||e_q||$ order $||e_q||$ order $||e_q||$ order
10 4.0632e-02 2.1357e-03 8.1067e-05 2.5683e-06
20 1.0377e-02 1.9692 2.6569e-04 3.0069 5.1201e-06 3.9849 8.0223e-08 5.0007
30 4.6494e-03 1.9801 7.9028e-05 2.9905 1.0154e-06 3.9902 1.0585e-08 4.9952
40 2.6263e-03 1.9854 3.3377e-05 2.9961 3.2190e-07 3.9933 2.5136e-09 4.9976
50 1.6852e-03 1.9884 1.7106e-05 2.9956 1.3200e-07 3.9949 8.2408e-10 4.9977
60 1.1723e-03 1.9905 9.9056e-06 2.9965 6.3705e-08 3.9959 3.3129e-10 4.9982
70 8.6237e-04 1.9918 6.2400e-06 2.9978 3.4405e-08 3.9965 1.5331e-10 4.9986
80 6.6088e-04 1.9929 4.1817e-06 2.9975 2.0176e-08 3.9969 7.8645e-11 4.9990
Table 3.  The $L^2$ errors $||e_r||$ and their orders of convergence for Example 1 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||e_r||$ order $||e_r||$ order $||e_r||$ order $||e_r||$ order
10 5.6489e-02 2.6049e-03 1.0690e-04 3.1989e-06
20 1.4417e-02 1.9702 3.1480e-04 3.0487 6.8578e-06 3.9624 9.3967e-08 5.0893
30 6.4714e-03 1.9755 9.2291e-05 3.0261 1.3672e-06 3.9772 1.2251e-08 5.0247
40 3.6555e-03 1.9854 3.8721e-05 3.0192 4.3448e-07 3.9849 2.8903e-09 5.0203
50 2.3462e-03 1.9872 1.9756e-05 3.0157 1.7839e-07 3.9893 9.4370e-10 5.0161
60 1.6322e-03 1.9903 1.1409e-05 3.0115 8.6173e-08 3.9908 3.7854e-10 5.0103
70 1.2006e-03 1.9923 7.1733e-06 3.0103 4.6565e-08 3.9929 1.7483e-10 5.0114
80 9.2012e-04 1.9926 4.8003e-06 3.0082 2.7319e-08 3.9936 8.9582e-11 5.0075
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p=1 p=2 p=3 p=4
N $||e_r||$ order $||e_r||$ order $||e_r||$ order $||e_r||$ order
10 5.6489e-02 2.6049e-03 1.0690e-04 3.1989e-06
20 1.4417e-02 1.9702 3.1480e-04 3.0487 6.8578e-06 3.9624 9.3967e-08 5.0893
30 6.4714e-03 1.9755 9.2291e-05 3.0261 1.3672e-06 3.9772 1.2251e-08 5.0247
40 3.6555e-03 1.9854 3.8721e-05 3.0192 4.3448e-07 3.9849 2.8903e-09 5.0203
50 2.3462e-03 1.9872 1.9756e-05 3.0157 1.7839e-07 3.9893 9.4370e-10 5.0161
60 1.6322e-03 1.9903 1.1409e-05 3.0115 8.6173e-08 3.9908 3.7854e-10 5.0103
70 1.2006e-03 1.9923 7.1733e-06 3.0103 4.6565e-08 3.9929 1.7483e-10 5.0114
80 9.2012e-04 1.9926 4.8003e-06 3.0082 2.7319e-08 3.9936 8.9582e-11 5.0075
Table 4.  The $L^2$ errors $||\bar{e}_u||$ and their orders of convergence for Example 1 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order
10 8.2164e-03 2.8426e-04 8.5926e-06 2.7880e-07
20 9.2680e-04 3.1482 1.4330e-05 4.3101 2.1723e-07 5.3058 2.5301e-09 6.7839
30 2.7332e-04 3.0116 2.7906e-06 4.0351 2.8408e-08 5.0172 2.2288e-10 5.9916
40 1.1432e-04 3.0299 8.4343e-07 4.1592 6.6969e-09 5.0230 3.4499e-11 6.4853
50 5.8315e-05 3.0166 3.4528e-07 4.0025 2.1868e-09 5.0156 9.0621e-12 5.9909
60 3.3724e-05 3.0038 1.6466e-07 4.0614 8.7923e-10 4.9975 2.9629e-12 6.1317
70 2.1167e-05 3.0215 8.7954e-08 4.0679 4.0603e-10 5.0121 1.1105e-12 6.3662
80 1.4176e-05 3.0022 5.1484e-08 4.0106 2.0829e-10 4.9988 4.9946e-13 5.9839
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p=1 p=2 p=3 p=4
N $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order
10 8.2164e-03 2.8426e-04 8.5926e-06 2.7880e-07
20 9.2680e-04 3.1482 1.4330e-05 4.3101 2.1723e-07 5.3058 2.5301e-09 6.7839
30 2.7332e-04 3.0116 2.7906e-06 4.0351 2.8408e-08 5.0172 2.2288e-10 5.9916
40 1.1432e-04 3.0299 8.4343e-07 4.1592 6.6969e-09 5.0230 3.4499e-11 6.4853
50 5.8315e-05 3.0166 3.4528e-07 4.0025 2.1868e-09 5.0156 9.0621e-12 5.9909
60 3.3724e-05 3.0038 1.6466e-07 4.0614 8.7923e-10 4.9975 2.9629e-12 6.1317
70 2.1167e-05 3.0215 8.7954e-08 4.0679 4.0603e-10 5.0121 1.1105e-12 6.3662
80 1.4176e-05 3.0022 5.1484e-08 4.0106 2.0829e-10 4.9988 4.9946e-13 5.9839
Table 5.  The $L^2$ errors $||e_u||$ and their orders of convergence for Example 2 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||e_u||$ order $||e_u||$ order $||e_u||$ order $||e_u||$ order
10 6.2562e-02 9.8763e-04 3.1041e-05 9.8565e-07
20 1.5553e-02 2.0081 1.0290e-04 3.2627 1.9775e-06 3.9724 3.0496e-08 5.0144
30 6.8955e-03 2.0061 3.0045e-05 3.0362 3.9293e-07 3.9854 4.0031e-09 5.0079
40 3.8742e-03 2.0040 1.2632e-05 3.0119 1.2468e-07 3.9901 9.4846e-10 5.0055
50 2.4781e-03 2.0025 6.4589e-06 3.0060 5.1154e-08 3.9925 3.1049e-10 5.0043
60 1.7203e-03 2.0021 3.7351e-06 3.0039 2.4696e-08 3.9940 1.2470e-10 5.0035
70 1.2636e-03 2.0014 2.3511e-06 3.0028 1.3341e-08 3.9948 5.7668e-11 5.0030
80 9.6732e-04 2.0009 1.5745e-06 3.0026 7.8247e-09 3.9957 2.9568e-11 5.0026
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p=1 p=2 p=3 p=4
N $||e_u||$ order $||e_u||$ order $||e_u||$ order $||e_u||$ order
10 6.2562e-02 9.8763e-04 3.1041e-05 9.8565e-07
20 1.5553e-02 2.0081 1.0290e-04 3.2627 1.9775e-06 3.9724 3.0496e-08 5.0144
30 6.8955e-03 2.0061 3.0045e-05 3.0362 3.9293e-07 3.9854 4.0031e-09 5.0079
40 3.8742e-03 2.0040 1.2632e-05 3.0119 1.2468e-07 3.9901 9.4846e-10 5.0055
50 2.4781e-03 2.0025 6.4589e-06 3.0060 5.1154e-08 3.9925 3.1049e-10 5.0043
60 1.7203e-03 2.0021 3.7351e-06 3.0039 2.4696e-08 3.9940 1.2470e-10 5.0035
70 1.2636e-03 2.0014 2.3511e-06 3.0028 1.3341e-08 3.9948 5.7668e-11 5.0030
80 9.6732e-04 2.0009 1.5745e-06 3.0026 7.8247e-09 3.9957 2.9568e-11 5.0026
Table 6.  The $L^2$ errors $||e_q||$ and their orders of convergence for Example 2 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||e_q||$ order $||e_q||$ order $||e_q||$ order $||e_q||$ order
10 6.9746e-02 5.7247e-03 4.8429e-04 3.8023e-05
20 1.7340e-02 2.0080 7.3031e-04 2.9706 3.0517e-05 3.9882 1.2060e-06 4.9786
30 7.6873e-03 2.0062 2.1756e-04 2.9867 6.0452e-06 3.9930 1.5947e-07 4.9898
40 4.3192e-03 2.0039 9.1951e-05 2.9936 1.9150e-06 3.9959 3.7902e-08 4.9946
50 2.7627e-03 2.0026 4.7116e-05 2.9965 7.8488e-07 3.9972 1.2429e-08 4.9967
60 1.9179e-03 2.0018 2.7278e-05 2.9976 3.7864e-07 3.9981 4.9968e-09 4.9980
70 1.4087e-03 2.0017 1.7182e-05 2.9985 2.0443e-07 3.9984 2.3124e-09 4.9984
80 1.0784e-03 2.0009 1.1512e-05 2.9991 1.1985e-07 3.9989 1.1862e-09 4.9991
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p=1 p=2 p=3 p=4
N $||e_q||$ order $||e_q||$ order $||e_q||$ order $||e_q||$ order
10 6.9746e-02 5.7247e-03 4.8429e-04 3.8023e-05
20 1.7340e-02 2.0080 7.3031e-04 2.9706 3.0517e-05 3.9882 1.2060e-06 4.9786
30 7.6873e-03 2.0062 2.1756e-04 2.9867 6.0452e-06 3.9930 1.5947e-07 4.9898
40 4.3192e-03 2.0039 9.1951e-05 2.9936 1.9150e-06 3.9959 3.7902e-08 4.9946
50 2.7627e-03 2.0026 4.7116e-05 2.9965 7.8488e-07 3.9972 1.2429e-08 4.9967
60 1.9179e-03 2.0018 2.7278e-05 2.9976 3.7864e-07 3.9981 4.9968e-09 4.9980
70 1.4087e-03 2.0017 1.7182e-05 2.9985 2.0443e-07 3.9984 2.3124e-09 4.9984
80 1.0784e-03 2.0009 1.1512e-05 2.9991 1.1985e-07 3.9989 1.1862e-09 4.9991
Table 7.  The $L^2$ errors $||e_r||$ and their orders of convergence for Example 2 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||e_r||$ order $||e_r||$ order $||e_r||$ order $||e_r||$ order
10 8.1901e-02 1.0758e-02 1.0971e-03 9.6872e-05
20 2.1870e-02 1.9049 1.3910e-03 2.9512 6.7331e-05 4.0263 3.2018e-06 4.9191
30 1.0250e-02 1.8691 4.1775e-04 2.9667 1.3681e-05 3.9303 4.2892e-07 4.9578
40 5.9921e-03 1.8661 1.7881e-04 2.9496 4.4205e-06 3.9271 1.0318e-07 4.9527
50 3.9450e-03 1.8732 9.2744e-05 2.9420 1.8385e-06 3.9316 3.4162e-08 4.9536
60 2.7990e-03 1.8823 5.4251e-05 2.9411 8.9679e-07 3.9375 1.3837e-08 4.9570
70 2.0912e-03 1.8912 3.4467e-05 2.9427 4.8834e-07 3.9430 6.4409e-09 4.9606
80 1.6227e-03 1.8995 2.3258e-05 2.9458 2.8825e-07 3.9480 3.3196e-09 4.9638
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p=1 p=2 p=3 p=4
N $||e_r||$ order $||e_r||$ order $||e_r||$ order $||e_r||$ order
10 8.1901e-02 1.0758e-02 1.0971e-03 9.6872e-05
20 2.1870e-02 1.9049 1.3910e-03 2.9512 6.7331e-05 4.0263 3.2018e-06 4.9191
30 1.0250e-02 1.8691 4.1775e-04 2.9667 1.3681e-05 3.9303 4.2892e-07 4.9578
40 5.9921e-03 1.8661 1.7881e-04 2.9496 4.4205e-06 3.9271 1.0318e-07 4.9527
50 3.9450e-03 1.8732 9.2744e-05 2.9420 1.8385e-06 3.9316 3.4162e-08 4.9536
60 2.7990e-03 1.8823 5.4251e-05 2.9411 8.9679e-07 3.9375 1.3837e-08 4.9570
70 2.0912e-03 1.8912 3.4467e-05 2.9427 4.8834e-07 3.9430 6.4409e-09 4.9606
80 1.6227e-03 1.8995 2.3258e-05 2.9458 2.8825e-07 3.9480 3.3196e-09 4.9638
Table 8.  The $L^2$ errors $||\bar{e}_u||$ and their orders of convergence for Example 2 on uniform meshes having $N = $ 10, 20, 30, 40, 50, 60, 70, 80 elements using $P^p$, $p = 1$ to 4.
p=1 p=2 p=3 p=4
N $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order
10 4.3648e-02 1.7020e-03 4.0015e-06 4.6584e-08
20 5.6286e-03 2.9551 5.4668e-05 4.9604 6.6219e-08 5.9172 7.1450e-10 6.0268
30 1.6836e-03 2.9766 7.3187e-06 4.9594 8.1013e-09 5.1816 6.2456e-11 6.0107
40 7.1340e-04 2.9847 1.7583e-06 4.9572 1.9000e-09 5.0409 1.1093e-11 6.0071
50 3.6619e-04 2.9886 5.8254e-07 4.9506 6.2116e-10 5.0103 2.9045e-12 6.0053
60 2.1226e-04 2.9911 2.3667e-07 4.9403 2.4956e-10 5.0015 9.7196e-13 6.0042
70 1.3382e-04 2.9926 1.1074e-07 4.9269 1.1549e-10 4.9985 3.9620e-13 5.8215
80 8.9728e-05 2.9934 5.7480e-08 4.9108 5.9257e-11 4.9973 1.8005e-13 5.9064
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p=1 p=2 p=3 p=4
N $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order $||\bar{e}_u||$ order
10 4.3648e-02 1.7020e-03 4.0015e-06 4.6584e-08
20 5.6286e-03 2.9551 5.4668e-05 4.9604 6.6219e-08 5.9172 7.1450e-10 6.0268
30 1.6836e-03 2.9766 7.3187e-06 4.9594 8.1013e-09 5.1816 6.2456e-11 6.0107
40 7.1340e-04 2.9847 1.7583e-06 4.9572 1.9000e-09 5.0409 1.1093e-11 6.0071
50 3.6619e-04 2.9886 5.8254e-07 4.9506 6.2116e-10 5.0103 2.9045e-12 6.0053
60 2.1226e-04 2.9911 2.3667e-07 4.9403 2.4956e-10 5.0015 9.7196e-13 6.0042
70 1.3382e-04 2.9926 1.1074e-07 4.9269 1.1549e-10 4.9985 3.9620e-13 5.8215
80 8.9728e-05 2.9934 5.7480e-08 4.9108 5.9257e-11 4.9973 1.8005e-13 5.9064
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