doi: 10.3934/cpaa.2021132
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Semi-discrete and fully discrete HDG methods for Burgers' equation

School of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author

Received  January 2021 Revised  June 2021 Early access July 2021

Fund Project: This work was supported by National Natural Science Foundation of China (11771312)

This paper proposes semi-discrete and fully discrete hybridizable discontinuous Galerkin (HDG) methods for the Burgers' equation in two and three dimensions. In the spatial discretization, we use piecewise polynomials of degrees $ k \ (k \geq 1), k-1 $ and $ l \ (l = k-1; k) $ to approximate the scalar function, flux variable and the interface trace of scalar function, respectively. In the full discretization method, we apply a backward Euler scheme for the temporal discretization. Optimal a priori error estimates are derived. Numerical experiments are presented to support the theoretical results.

Citation: Zimo Zhu, Gang Chen, Xiaoping Xie. Semi-discrete and fully discrete HDG methods for Burgers' equation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021132
References:
[1]

E. N. Aksan, A numerical solution of Burgers' equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput., 170 (2005), 895-904.  doi: 10.1016/j.amc.2004.12.027.  Google Scholar

[2]

R. Alexande, Diagonally Implicit Runge-Kutta Methods for Stiff O.D.E.'s, SIAM J. Numer. Anal., 14 (1977), 1006-1021.  doi: 10.1137/0714068.  Google Scholar

[3]

A. AliG. Gardner and L. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Method. Appl. M., 100 (1992), 325-337.  doi: 10.1016/0045-7825(92)90088-2.  Google Scholar

[4]

P. Arminjon and C. Beauchamp, A finite element method for Burgers' equation in hydrodynamics, Int. J. num. Meth. Eng, 12 (1978), 415-428.  doi: 10.1002/nme.1620120304.  Google Scholar

[5]

P. Arminjon and C. Beauchamp, Continuous and discontinuous finite element methods for Burgers' equation, Comput. Methods Appl. M., 25 (1981), 65-84.  doi: 10.1016/0045-7825(81)90069-4.  Google Scholar

[6]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[7]

J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), 171-199.  doi: 10.1016/S0065-2156(08)70100-5.  Google Scholar

[8]

J. Burgers, Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion, Springer, Dordrecht, 1995. doi: 10.1007/978-94-011-0195-0_10.  Google Scholar

[9]

E. Burman, Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation, Math. Mod. Meth. Appl. S., 25 (2015), 2015-2042.  doi: 10.1142/S0218202515500517.  Google Scholar

[10]

J. CaldwellP. Wanless and A. Cook, A finite element approach to Burgers' equation, Appl. Math. Model., 5 (1981), 189-193.  doi: 10.1016/0307-904X(81)90043-3.  Google Scholar

[11]

G. Chen and X. Xie, A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses, Comput. Meth. Appl. Mat., 16 (2016), 389-408.  doi: 10.1515/cmam-2016-0012.  Google Scholar

[12]

H. Chen and Z. Jiang, A characteristics-mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 15 (2004), 29-51.  doi: 10.1007/BF02935745.  Google Scholar

[13]

H. ChenP. Lu and X. Xu, A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Comput. Phys., 264 (2014), 133-151.  doi: 10.1016/j.jcp.2014.01.042.  Google Scholar

[14]

Y. Chen and T. Zhang, A weak Galerkin finite element method for Burgers' equation, J. Comput. Appl. Math., 348 (2019), 103-109.  doi: 10.1016/j.cam.2018.08.044.  Google Scholar

[15]

B. CockburnJ. Gopalakrishnan and N. Nguyen, Analysis of HDG methods for Stokes flow, Math. Comput., 80 (2011), 723-760.  doi: 10.1090/S0025-5718-2010-02410-X.  Google Scholar

[16]

A. Dogan, A Galerkin finite element approach to Burgers' equation, Appl. Math. Comput., 157 (2004), 331-346.  doi: 10.1016/j.amc.2003.08.037.  Google Scholar

[17]

R. Guzzi and L. Stefanutti, The Role of Airflow in Airborne Transmission of COVID 19, Int. J. Biol. Sci., 4 (2021), 121-131.  doi: 10.13133/2532-5876/17224.  Google Scholar

[18]

Y. Han, H. Chen, X. Wang and X. Xie, EXtended HDG methods for second order elliptic interface problems, J. Sci. Comput., 84 (2020), 22. doi: 10.1007/s10915-020-01272-3.  Google Scholar

[19]

J. Heywood and R. Rannacher, Finite-Element Approximation of the Nonstationary Navier-Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.  Google Scholar

[20]

X. HuP. Huang and X. Feng, Two-Grid Method for Burgers' Equation by a New Mixed Finite Element Scheme, Math. Model. Anal., 19 (2014), 1-17.  doi: 10.3846/13926292.2014.892902.  Google Scholar

[21]

X. HuP. Huang and X. Feng, A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation, Appl. Math-Czech., 61 (2016), 27-45.  doi: 10.1007/s10492-016-0120-3.  Google Scholar

[22]

A. Hussein and H. Kashkool, Weak Galerkin finite element method for solving one-dimensional coupled Burgers' equations, J. Appl. Math. Comput., 63 (2020), 265-293.  doi: 10.1007/s12190-020-01317-8.  Google Scholar

[23]

O. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second order elliptic problems, SIAM J. Numer. Anal., 45 (2007), 641-665.  doi: 10.1137/05063979X.  Google Scholar

[24]

S. KutluayA. Esen and I. Dag, Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167 (2004), 21-33.  doi: 10.1016/j.cam.2003.09.043.  Google Scholar

[25]

B. Li and X. Xie, Analysis of a family of HDG methods for second order elliptic problems, J. Comput. Appl. Math., 307 (2016), 37-51.  doi: 10.1016/j.cam.2016.04.027.  Google Scholar

[26]

C. W. Lucchi, Improvement of MacCormack's scheme for Burgers' equation. Using a finite element method, Int. J. num. Meth. Eng, 15 (1980), 537-555.  doi: 10.1002/nme.1620150406.  Google Scholar

[27]

R. Mittal and A. Tripathi, Numerical solutions of two-dimensional Burgers' equations using modified Bi-cubic B-spline finite elements, Eng. Comput., 32 (2015), 1275-1306.  doi: 10.1108/EC-04-2014-0067.  Google Scholar

[28]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228 (2009), 3232-3254.  doi: 10.1016/j.jcp.2009.01.030.  Google Scholar

[29]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228 (2009), 8841-8855.  doi: 10.1016/j.jcp.2009.08.030.  Google Scholar

[30]

T. ÖzişE. Aksan and A. Özdeş, A finite element approach for solution of Burgers' equation, Appl. Math. Comput., 139 (2003), 417-428.  doi: 10.1016/S0096-3003(02)00204-7.  Google Scholar

[31]

A. PanyN. Nataraj and S. Singh, A new mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 23 (2007), 43-55.  doi: 10.1007/BF02831957.  Google Scholar

[32]

W. QiuJ. Shen and K. Shi, An HDG method for linear elasticity with strong symmetric stresses, Math. Comput., 87 (2016), 69-93.  doi: 10.1090/mcom/3249.  Google Scholar

[33]

L. ShaoX. Feng and Y. He, The local discontinuous Galerkin finite element method for Burger's equation, Math. Comput. Model., 54 (2011), 2943-2954.  doi: 10.1016/j.mcm.2011.07.016.  Google Scholar

[34]

D. ShiJ. Zhou and D. Shi, A new low order least squares nonconforming characteristics mixed finite element method for Burgers' equation, Appl. Math. Comput., 219 (2013), 11302-11310.  doi: 10.1016/j.amc.2013.05.037.  Google Scholar

[35] Z. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013.   Google Scholar
[36]

H. SterckT. ManteuffelS. Mccormick and L. Olson, Numerical conservation properities of H(div)-conforming least-squares finite element methods for the Burgers equation, SIAM J. Sci. Comput., 26 (2005), 1573-1597.  doi: 10.1137/S1064827503430758.  Google Scholar

[37]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[38]

Y. UçarN. Yaǧmurlu and İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Meth. Part. D. E., 35 (2019), 478-492.  doi: 10.1002/num.22309.  Google Scholar

[39]

J. Warga, Optimal Control of Differential and Functional Equations, 1st edition, Academic Press, New York, 1972. doi: 10.1016/C2013-0-11669-8.  Google Scholar

[40]

D. Winterscheidt and K. Surana, p-version least-squares finite element formulation of Burgers' equation, Int. J. num. Meth. Eng, 36 (2010), 3629-3646.  doi: 10.1002/nme.1620362105.  Google Scholar

[41]

G. ZhaoX. Yu and R. Zhang, The new numerical method for solving the system of two-dimensional Burgers' equations, Comput. Math. Appl., 62 (2011), 3279-3291.  doi: 10.1016/j.camwa.2011.08.044.  Google Scholar

show all references

References:
[1]

E. N. Aksan, A numerical solution of Burgers' equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput., 170 (2005), 895-904.  doi: 10.1016/j.amc.2004.12.027.  Google Scholar

[2]

R. Alexande, Diagonally Implicit Runge-Kutta Methods for Stiff O.D.E.'s, SIAM J. Numer. Anal., 14 (1977), 1006-1021.  doi: 10.1137/0714068.  Google Scholar

[3]

A. AliG. Gardner and L. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Method. Appl. M., 100 (1992), 325-337.  doi: 10.1016/0045-7825(92)90088-2.  Google Scholar

[4]

P. Arminjon and C. Beauchamp, A finite element method for Burgers' equation in hydrodynamics, Int. J. num. Meth. Eng, 12 (1978), 415-428.  doi: 10.1002/nme.1620120304.  Google Scholar

[5]

P. Arminjon and C. Beauchamp, Continuous and discontinuous finite element methods for Burgers' equation, Comput. Methods Appl. M., 25 (1981), 65-84.  doi: 10.1016/0045-7825(81)90069-4.  Google Scholar

[6]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[7]

J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), 171-199.  doi: 10.1016/S0065-2156(08)70100-5.  Google Scholar

[8]

J. Burgers, Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion, Springer, Dordrecht, 1995. doi: 10.1007/978-94-011-0195-0_10.  Google Scholar

[9]

E. Burman, Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation, Math. Mod. Meth. Appl. S., 25 (2015), 2015-2042.  doi: 10.1142/S0218202515500517.  Google Scholar

[10]

J. CaldwellP. Wanless and A. Cook, A finite element approach to Burgers' equation, Appl. Math. Model., 5 (1981), 189-193.  doi: 10.1016/0307-904X(81)90043-3.  Google Scholar

[11]

G. Chen and X. Xie, A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses, Comput. Meth. Appl. Mat., 16 (2016), 389-408.  doi: 10.1515/cmam-2016-0012.  Google Scholar

[12]

H. Chen and Z. Jiang, A characteristics-mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 15 (2004), 29-51.  doi: 10.1007/BF02935745.  Google Scholar

[13]

H. ChenP. Lu and X. Xu, A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Comput. Phys., 264 (2014), 133-151.  doi: 10.1016/j.jcp.2014.01.042.  Google Scholar

[14]

Y. Chen and T. Zhang, A weak Galerkin finite element method for Burgers' equation, J. Comput. Appl. Math., 348 (2019), 103-109.  doi: 10.1016/j.cam.2018.08.044.  Google Scholar

[15]

B. CockburnJ. Gopalakrishnan and N. Nguyen, Analysis of HDG methods for Stokes flow, Math. Comput., 80 (2011), 723-760.  doi: 10.1090/S0025-5718-2010-02410-X.  Google Scholar

[16]

A. Dogan, A Galerkin finite element approach to Burgers' equation, Appl. Math. Comput., 157 (2004), 331-346.  doi: 10.1016/j.amc.2003.08.037.  Google Scholar

[17]

R. Guzzi and L. Stefanutti, The Role of Airflow in Airborne Transmission of COVID 19, Int. J. Biol. Sci., 4 (2021), 121-131.  doi: 10.13133/2532-5876/17224.  Google Scholar

[18]

Y. Han, H. Chen, X. Wang and X. Xie, EXtended HDG methods for second order elliptic interface problems, J. Sci. Comput., 84 (2020), 22. doi: 10.1007/s10915-020-01272-3.  Google Scholar

[19]

J. Heywood and R. Rannacher, Finite-Element Approximation of the Nonstationary Navier-Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.  Google Scholar

[20]

X. HuP. Huang and X. Feng, Two-Grid Method for Burgers' Equation by a New Mixed Finite Element Scheme, Math. Model. Anal., 19 (2014), 1-17.  doi: 10.3846/13926292.2014.892902.  Google Scholar

[21]

X. HuP. Huang and X. Feng, A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation, Appl. Math-Czech., 61 (2016), 27-45.  doi: 10.1007/s10492-016-0120-3.  Google Scholar

[22]

A. Hussein and H. Kashkool, Weak Galerkin finite element method for solving one-dimensional coupled Burgers' equations, J. Appl. Math. Comput., 63 (2020), 265-293.  doi: 10.1007/s12190-020-01317-8.  Google Scholar

[23]

O. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second order elliptic problems, SIAM J. Numer. Anal., 45 (2007), 641-665.  doi: 10.1137/05063979X.  Google Scholar

[24]

S. KutluayA. Esen and I. Dag, Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167 (2004), 21-33.  doi: 10.1016/j.cam.2003.09.043.  Google Scholar

[25]

B. Li and X. Xie, Analysis of a family of HDG methods for second order elliptic problems, J. Comput. Appl. Math., 307 (2016), 37-51.  doi: 10.1016/j.cam.2016.04.027.  Google Scholar

[26]

C. W. Lucchi, Improvement of MacCormack's scheme for Burgers' equation. Using a finite element method, Int. J. num. Meth. Eng, 15 (1980), 537-555.  doi: 10.1002/nme.1620150406.  Google Scholar

[27]

R. Mittal and A. Tripathi, Numerical solutions of two-dimensional Burgers' equations using modified Bi-cubic B-spline finite elements, Eng. Comput., 32 (2015), 1275-1306.  doi: 10.1108/EC-04-2014-0067.  Google Scholar

[28]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228 (2009), 3232-3254.  doi: 10.1016/j.jcp.2009.01.030.  Google Scholar

[29]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228 (2009), 8841-8855.  doi: 10.1016/j.jcp.2009.08.030.  Google Scholar

[30]

T. ÖzişE. Aksan and A. Özdeş, A finite element approach for solution of Burgers' equation, Appl. Math. Comput., 139 (2003), 417-428.  doi: 10.1016/S0096-3003(02)00204-7.  Google Scholar

[31]

A. PanyN. Nataraj and S. Singh, A new mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 23 (2007), 43-55.  doi: 10.1007/BF02831957.  Google Scholar

[32]

W. QiuJ. Shen and K. Shi, An HDG method for linear elasticity with strong symmetric stresses, Math. Comput., 87 (2016), 69-93.  doi: 10.1090/mcom/3249.  Google Scholar

[33]

L. ShaoX. Feng and Y. He, The local discontinuous Galerkin finite element method for Burger's equation, Math. Comput. Model., 54 (2011), 2943-2954.  doi: 10.1016/j.mcm.2011.07.016.  Google Scholar

[34]

D. ShiJ. Zhou and D. Shi, A new low order least squares nonconforming characteristics mixed finite element method for Burgers' equation, Appl. Math. Comput., 219 (2013), 11302-11310.  doi: 10.1016/j.amc.2013.05.037.  Google Scholar

[35] Z. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013.   Google Scholar
[36]

H. SterckT. ManteuffelS. Mccormick and L. Olson, Numerical conservation properities of H(div)-conforming least-squares finite element methods for the Burgers equation, SIAM J. Sci. Comput., 26 (2005), 1573-1597.  doi: 10.1137/S1064827503430758.  Google Scholar

[37]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[38]

Y. UçarN. Yaǧmurlu and İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Meth. Part. D. E., 35 (2019), 478-492.  doi: 10.1002/num.22309.  Google Scholar

[39]

J. Warga, Optimal Control of Differential and Functional Equations, 1st edition, Academic Press, New York, 1972. doi: 10.1016/C2013-0-11669-8.  Google Scholar

[40]

D. Winterscheidt and K. Surana, p-version least-squares finite element formulation of Burgers' equation, Int. J. num. Meth. Eng, 36 (2010), 3629-3646.  doi: 10.1002/nme.1620362105.  Google Scholar

[41]

G. ZhaoX. Yu and R. Zhang, The new numerical method for solving the system of two-dimensional Burgers' equations, Comput. Math. Appl., 62 (2011), 3279-3291.  doi: 10.1016/j.camwa.2011.08.044.  Google Scholar

Figure 1.  The domain : $ 4\times 4 $ (left) and $ 8\times 8 $ (right) meshes
Figure 2.  The domain: $ 2\times 2\times 2 $(left) and $ 4\times 4\times 4 $(right) meshes
Table 1.  History of convergence for Example 5.1 with $ \nu = 1, k = 1 $
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 2.1597e-01 2.9311e-01
$ 8\times 8 $ 5.4132e-02 2.00 1.4864e-01 0.98
$ 16\times 16 $ 1.3543e-02 2.00 7.4578e-02 0.99
$ 32\times 32 $ 3.3865e-03 2.00 3.7322e-02 1.00
$ 64\times 64 $ 8.4665e-04 2.00 1.8665e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $2.4145e-01-3.1279e-01-
$ 8\times 8 $6.0180e-022.001.5806e-010.98
$ 16\times 16 $1.5038e-022.007.9255e-021.00
$32\times 32$3.7593e-032.003.9656e-021.00
$64\times 64$9.3980e-042.001.9832e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 2.1597e-01 2.9311e-01
$ 8\times 8 $ 5.4132e-02 2.00 1.4864e-01 0.98
$ 16\times 16 $ 1.3543e-02 2.00 7.4578e-02 0.99
$ 32\times 32 $ 3.3865e-03 2.00 3.7322e-02 1.00
$ 64\times 64 $ 8.4665e-04 2.00 1.8665e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $2.4145e-01-3.1279e-01-
$ 8\times 8 $6.0180e-022.001.5806e-010.98
$ 16\times 16 $1.5038e-022.007.9255e-021.00
$32\times 32$3.7593e-032.003.9656e-021.00
$64\times 64$9.3980e-042.001.9832e-021.00
Table 2.  History of convergence for Example 5.1 with $ \nu = 0.01, k = 1 $
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.1207e-01 - 2.9063e-01 -
$ 8\times 8 $ 3.3444e-02 1.74 1.4978e-01 0.96
$ 16\times 16 $ 8.5650e-03 1.97 7.4851e-02 1.00
$ 32\times 32 $ 2.1460e-03 2.00 3.7359e-02 1.00
$ 64\times 64 $ 5.3674e-04 2.00 1.8669e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.3157e-01-3.0275e-01-
$ 8\times 8 $4.1890e-021.651.6350e-010.89
$ 16\times 16 $1.0394e-022.018.0699e-021.02
$32\times 32$2.5616e-032.023.9968e-021.01
$64\times 64$6.3806e-042.011.9936e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.1207e-01 - 2.9063e-01 -
$ 8\times 8 $ 3.3444e-02 1.74 1.4978e-01 0.96
$ 16\times 16 $ 8.5650e-03 1.97 7.4851e-02 1.00
$ 32\times 32 $ 2.1460e-03 2.00 3.7359e-02 1.00
$ 64\times 64 $ 5.3674e-04 2.00 1.8669e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.3157e-01-3.0275e-01-
$ 8\times 8 $4.1890e-021.651.6350e-010.89
$ 16\times 16 $1.0394e-022.018.0699e-021.02
$32\times 32$2.5616e-032.023.9968e-021.01
$64\times 64$6.3806e-042.011.9936e-021.00
Table 3.  History of convergence for Example 5.1 with $ \nu = 1, k = 2 $
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.3700e-02 - 4.1763e-02 -
$ 8\times 8 $ 1.5937e-03 3.10 1.0597e-02 1.98
$ 16\times 16 $ 1.9284e-04 3.05 2.6642e-03 1.99
$ 32\times 32 $ 2.3736e-05 3.02 6.6756e-04 2.00
$ 64\times 64 $ 2.9449e-06 3.01 1.6705e-04 2.00
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.4714e-02-4.3007e-02-
$ 8\times 8 $1.7267e-033.091.0913e-021.98
$ 16\times 16 $2.1016e-043.042.7449e-031.99
$ 32\times 32 $2.5953e-053.026.8800e-042.00
$ 64\times 64 $3.2256e-063.011.7220e-042.00
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.3700e-02 - 4.1763e-02 -
$ 8\times 8 $ 1.5937e-03 3.10 1.0597e-02 1.98
$ 16\times 16 $ 1.9284e-04 3.05 2.6642e-03 1.99
$ 32\times 32 $ 2.3736e-05 3.02 6.6756e-04 2.00
$ 64\times 64 $ 2.9449e-06 3.01 1.6705e-04 2.00
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.4714e-02-4.3007e-02-
$ 8\times 8 $1.7267e-033.091.0913e-021.98
$ 16\times 16 $2.1016e-043.042.7449e-031.99
$ 32\times 32 $2.5953e-053.026.8800e-042.00
$ 64\times 64 $3.2256e-063.011.7220e-042.00
Table 4.  History of convergence for Example 5.1 with $ \nu = 0.01, k = 2 $
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 9.4294e-02 1.0218e-01
$ 8\times 8 $ 1.2210e-02 2.95 1.6179e-02 2.66
$ 16\times 16 $ 1.5272e-03 3.00 3.0797e-03 2.39
$ 32\times 32 $ 1.9068e-04 3.00 6.9507e-04 2.15
$ 64\times 64 $ 2.3817e-05 3.00 1.6880e-04 2.04
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $9.4922e-021.0298e-01
$ 8\times 8 $1.2259e-022.951.6577e-022.64
$ 16\times 16 $1.5310e-033.003.1770e-032.38
$ 32\times 32 $1.9114e-043.007.2041e-042.14
$ 64\times 64 $2.3877e-053.001.7525e-042.04
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 9.4294e-02 1.0218e-01
$ 8\times 8 $ 1.2210e-02 2.95 1.6179e-02 2.66
$ 16\times 16 $ 1.5272e-03 3.00 3.0797e-03 2.39
$ 32\times 32 $ 1.9068e-04 3.00 6.9507e-04 2.15
$ 64\times 64 $ 2.3817e-05 3.00 1.6880e-04 2.04
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $9.4922e-021.0298e-01
$ 8\times 8 $1.2259e-022.951.6577e-022.64
$ 16\times 16 $1.5310e-033.003.1770e-032.38
$ 32\times 32 $1.9114e-043.007.2041e-042.14
$ 64\times 64 $2.3877e-053.001.7525e-042.04
Table 5.  History of convergence with $ k = 3 $: Example 5.2
$ \Delta t $ HDG-I$ (l=3) $ HDG-II$ (l=2) $
$ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $
error order error order
0.2 2.2145e-03 2.2145e-03
0.1 3.7353e-04 2.57 3.7353e-04 2.57
0.05 5.7074e-05 2.71 5.7074e-05 2.71
0.025 8.1013e-06 2.82 8.1013e-06 2.82
0.0125 1.0947e-06 2.89 1.0947e-06 2.89
$ \Delta t $ HDG-I$ (l=3) $ HDG-II$ (l=2) $
$ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $
error order error order
0.2 2.2145e-03 2.2145e-03
0.1 3.7353e-04 2.57 3.7353e-04 2.57
0.05 5.7074e-05 2.71 5.7074e-05 2.71
0.025 8.1013e-06 2.82 8.1013e-06 2.82
0.0125 1.0947e-06 2.89 1.0947e-06 2.89
Table 6.  History of convergence with $ k = 1 $: Example 5.2
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 1.5445e-01 4.2222e-01
$ 16\times 16 $ 4.2801e-02 1.85 1.9709e-01 1.10
$ 32\times 32 $ 1.0908e-02 1.97 9.8335e-02 1.00
$ 64\times 64 $ 2.7412e-03 1.99 4.9190e-02 1.00
$ 128\times 128 $ 6.8606e-04 2.00 2.4598e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $1.7900e-014.5177e-01
$ 16\times 16 $4.7423e-021.922.0942e-011.11
$ 32\times 32 $1.1971e-021.991.0380e-011.01
$ 64\times 64 $3.0019e-032.005.1851e-021.00
$ 128\times 128 $7.5102e-042.002.5920e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 1.5445e-01 4.2222e-01
$ 16\times 16 $ 4.2801e-02 1.85 1.9709e-01 1.10
$ 32\times 32 $ 1.0908e-02 1.97 9.8335e-02 1.00
$ 64\times 64 $ 2.7412e-03 1.99 4.9190e-02 1.00
$ 128\times 128 $ 6.8606e-04 2.00 2.4598e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $1.7900e-014.5177e-01
$ 16\times 16 $4.7423e-021.922.0942e-011.11
$ 32\times 32 $1.1971e-021.991.0380e-011.01
$ 64\times 64 $3.0019e-032.005.1851e-021.00
$ 128\times 128 $7.5102e-042.002.5920e-021.00
Table 7.  History of convergence with $ k = 2 $: Example 5.2
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 2.8115e-02 4.9545e-02
$ 16\times 16 $ 4.7515e-03 2.56 2.1126e-02 1.23
$ 32\times 32 $ 6.1910e-04 2.94 5.6085e-03 1.91
$ 64\times 64 $ 7.7864e-05 2.99 1.4177e-03 1.98
$ 128\times 128 $ 9.7479e-06 3.00 3.5571e-04 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $2.9126e-025.0053e-02
$ 16\times 16 $4.8563e-032.582.1191e-021.24
$ 32\times 32 $6.3425e-042.945.6117e-031.92
$ 64\times 64 $7.9961e-052.991.4199e-031.98
$ 128\times 128 $1.0021e-053.003.5663e-041.99
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 2.8115e-02 4.9545e-02
$ 16\times 16 $ 4.7515e-03 2.56 2.1126e-02 1.23
$ 32\times 32 $ 6.1910e-04 2.94 5.6085e-03 1.91
$ 64\times 64 $ 7.7864e-05 2.99 1.4177e-03 1.98
$ 128\times 128 $ 9.7479e-06 3.00 3.5571e-04 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $2.9126e-025.0053e-02
$ 16\times 16 $4.8563e-032.582.1191e-021.24
$ 32\times 32 $6.3425e-042.945.6117e-031.92
$ 64\times 64 $7.9961e-052.991.4199e-031.98
$ 128\times 128 $1.0021e-053.003.5663e-041.99
Table 8.  History of convergence with $ k = 1 $: Example 5.3
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 6.9815e-01 5.6066e-01
$ 4\times 4\times 4 $ 1.7672e-01 1.98 3.0285e-01 0.89
$ 8\times 8\times 8 $ 4.4207e-02 2.00 1.5438e-01 0.97
$ 16\times 16\times 16 $ 1.1050e-02 2.00 7.7566e-02 0.99
$ 32\times 32\times 32 $ 2.7621e-03 2.00 3.8830e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $8.8064e-016.1105e-01
$ 4\times 4\times 4 $2.0917e-012.073.1971e-010.93
$ 8\times 8\times 8 $5.1528e-022.021.6186e-010.98
$ 16\times 16\times 16 $1.2835e-022.008.1193e-020.99
$ 32\times 32\times 32 $3.2055e-032.004.0630e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 6.9815e-01 5.6066e-01
$ 4\times 4\times 4 $ 1.7672e-01 1.98 3.0285e-01 0.89
$ 8\times 8\times 8 $ 4.4207e-02 2.00 1.5438e-01 0.97
$ 16\times 16\times 16 $ 1.1050e-02 2.00 7.7566e-02 0.99
$ 32\times 32\times 32 $ 2.7621e-03 2.00 3.8830e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $8.8064e-016.1105e-01
$ 4\times 4\times 4 $2.0917e-012.073.1971e-010.93
$ 8\times 8\times 8 $5.1528e-022.021.6186e-010.98
$ 16\times 16\times 16 $1.2835e-022.008.1193e-020.99
$ 32\times 32\times 32 $3.2055e-032.004.0630e-021.00
Table 9.  History of convergence with $ k = 2 $: Example 5.3
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 1.3095e-01 1.9090e-01
$ 4\times 4\times 4 $ 1.4825e-02 3.14 5.3052e-02 1.85
$ 8\times 8\times 8 $ 1.7531e-03 3.08 1.3659e-02 1.96
$ 16\times 16\times 16 $ 2.1463e-04 3.03 3.4459e-03 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $1.4705e-011.9582e-01
$ 4\times 4\times 4 $1.6034e-023.205.3962e-021.86
$ 8\times 8\times 8 $1.8868e-033.091.3871e-021.96
$ 16\times 16\times 16 $2.3104e-043.033.4979e-031.99
$ 32\times 32\times 32 $2.8681e-053.018.7713e-042.00
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 1.3095e-01 1.9090e-01
$ 4\times 4\times 4 $ 1.4825e-02 3.14 5.3052e-02 1.85
$ 8\times 8\times 8 $ 1.7531e-03 3.08 1.3659e-02 1.96
$ 16\times 16\times 16 $ 2.1463e-04 3.03 3.4459e-03 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $1.4705e-011.9582e-01
$ 4\times 4\times 4 $1.6034e-023.205.3962e-021.86
$ 8\times 8\times 8 $1.8868e-033.091.3871e-021.96
$ 16\times 16\times 16 $2.3104e-043.033.4979e-031.99
$ 32\times 32\times 32 $2.8681e-053.018.7713e-042.00
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