# American Institute of Mathematical Sciences

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

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

Received  March 2017 Revised  January 2018 Published  January 2019 Early access  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.

Citation: Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978. [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. [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. [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. [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. [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. [25] T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2010. [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. [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. [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. [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. [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. [31] S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM Journal on Numerical Analysis, 21 (1984), 217-235.  doi: 10.1137/0721016. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978. [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. [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. [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. [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. [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. [25] T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2010. [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. [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. [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. [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. [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. [31] S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM Journal on Numerical Analysis, 21 (1984), 217-235.  doi: 10.1137/0721016. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
Space-time graphs of the exact solution u (left) and the LDG solution uh (right) for Example 105 using N = 80 and p = 3.
Space-time graphs of the exact solution u (left) and the LDG solution uh (right) for Example 4 using N = 80 and p = 3.
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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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