# 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  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 & 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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