Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes

doi:10.1007/s42967-023-00330-5

Abstract

Abstract: A novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation on rectangular meshes. This CDG method with discontinuous P_k (k ≥ 1) elements converges to the true solution two orders above the continuous finite element counterpart. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the L² norm. A local post-process is defined which lifts a P_k CDG solution to a discontinuous P_k+2 solution. It is proved that the lifted P_k+2 solution converges at the optimal order. The numerical tests illustrate the theoretic findings.

Key words: Finite element, Conforming discontinuous Galerkin (CDG) method, Stabilizer free, Rectangular mesh, Superconvergent

CLC Number:

65N15
65N30

Xiu Ye, Shangyou Zhang. Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 411-425.

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References

1. Al-Twaeel, A., Hussian, S., Wang, X.: A stabilizer free weak Galerkin finite element method for parabolic equation. J. Comput. Appl. Math. 392, 113373 (2021)
2. Al-Taweel, A., Wang, X., Ye, X., Zhang, S.: A stabilizer free weak Galerkin method with supercloseness of order two. Numer. Methods Partial Differential Equations 37, 1012–1029 (2021)
3. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)
4. Brezzi, F., Fortin, M.: Mixed Hybrid Finite Elements. Springer-Verlag, New York (1991)
5. Chen, G., Feng, M., Xie, X.: A robust WG finite element method for convection-diffusion-reaction equations. J. Comput. Appl. Math. 315, 107–125 (2017)
6. Chen, W., Wang, F., Wang, Y.: Weak Galerkin method for the coupled Darcy-Stokes flow. IMA J. Numer. Anal. 36, 897–921 (2016)
7. Cui, M., Zhang, S.: On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation. J. Sci. Comput. 82, 5–15 (2020)
8. Deka, B., Roy, P.: Weak Galerkin finite element methods for parabolic interface problems with nonhomogeneous jump conditions. Numer. Funct. Anal. Optim. 40, 250–279 (2019)
9. Feng, Y., Liu, Y., Wang, R., Zhang, S.: A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Res. Archive 29, 2375–2389 (2021)
10. Gao, F., Ye, X., Zhang, S.: A discontinuous Galerkin finite element method without interior penalty terms. Adv. Appl. Math. Mech. 14(2), 299–314 (2022)
11. Gao, F., Zhang, S., Zhu, P.: Modified weak Galerkin method with weakly imposed boundary condition for convection-dominated diffusion equations. Appl. Numer. Math. 157, 490–504 (2020)
12. Guan, Q., Gunzburger, M., Zhao, W.: Weak-Galerkin finite element methods for a second-order elliptic variational inequality. Comput. Methods Appl. Mech. Engrg. 337, 677–688 (2018)
13. Hu, Q., He, Y., Wang, K.: Weak Galerkin method for the Helmholtz equation with DTN boundary condition. Int. J. Numer. Anal. Model. 17, 643–661 (2020)
14. Hu, X., Mu, L., Ye, X.: A weak Galerkin finite element method for the Navier-Stokes equations on polytopal meshes. J. Comput. Appl. Math. 362, 614–625 (2019)
15. Huang, W., Wang, Y.: Discrete maximum principle for the weak Galerkin method for anisotropic diffusion problems. Comm. Comput. Phys. 18, 65–90 (2015)
16. Li, G., Chen, Y., Huang, Y.: A new weak Galerkin finite element scheme for general second-order elliptic problems. J. Comput. Appl. Math. 344, 701–715 (2018)
17. Li, H., Mu, L., Ye, X.: Interior energy estimates for the weak Galerkin finite element method. Numer. Math. 139, 447–478 (2018)
18. Li, J., Ye, X., Zhang, S.: A weak Galerkin least-squares finite element method for div-curl systems. J. Comput. Phys. 363, 79–86 (2018)
19. Lin, R., Ye, X., Zhang, S., Zhu, P.: A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems. SIAM J. Numer. Anal. 56, 1482–1497 (2018)
20. Liu, J., Tavener, S., Wang, Z.: Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes. SIAM J. Sci. Comput. 40, 1229–1252 (2018)
21. Liu, J., Tavener, S., Wang, Z.: The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes. J. Comput. Phys. 359, 317330 (2018)
22. Liu, X., Li, J., Chen, Z.: A weak Galerkin finite element method for the Oseen equations. Adv. Comput. Math. 42, 1473–1490 (2016)
23. Mu, L., Ye, X., Zhang, S.: A stabilizer free, pressure robust and superconvergence weak Galerkin finite element method for the Stokes equations on polytopal mesh. SIAM. J. Sci. Comput. 43, A2614–A2637 (2021)
24. Mu, L., Ye, X., Zhang, S.: Development of pressure-robust discontinuous Galerkin finite element methods for the Stokes problem. J. Sci. Comput. 89(1), Paper No. 26, 25 (2021)
25. Peng, H., Zhai, Q., Zhang, R., Zhang, S.: Weak Galerkin and continuous Galerkin coupled finite element methods for the Stokes-Darcy interface problem. Commun. Comput. Phys. 28(3), 1147–1175 (2020)
26. Qi, W., Song, L.: Weak Galerkin method with implicit θ-schemes for second-order parabolic problems. Appl. Math. Comput. 336, 124731 (2020)
27. Shields, S., Li, J., Machorro, E.A.: Weak Galerkin methods for time-dependent Maxwell’s equations. Comput. Math. Appl. 74, 2106–2124 (2017)
28. Sun, M., Rui, H.: A coupling of weak Galerkin and mixed finite element methods for poroelasticity. Comput. Math. Appl. 73, 804–823 (2017)
29. Toprakseven, S.: A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients. Appl. Numer. Math. 168, 1–12 (2021)
30. Wang, C., Wang, J.: Discretization of div-curl systems by weak Galerkin finite element methods on polyhedral partitions. J. Sci. Comput. 68, 1144–1171 (2016)
31. Wang, C., Wang, J., Zhang, S.: Weak Galerkin finite element methods for quad-curl problems. J. Comput. Appl. Math. 428, Paper No. 115186, 13 (2023)
32. Wang, C., Zhang, S.: A weak Galerkin method for elasticity interface problems. J. Comput. Appl. Math. 419, Paper No. 114726, 14 (2023)
33. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)
34. Wang, J., Ye, X., Zhang, S.: A time-explicit weak Galerkin scheme for parabolic equations on polytopal partitions. J. Numer. Math. 31(2), 125–135 (2023)
35. Wang, J., Zhai, Q., Zhang, R., Zhang, S.: A weak Galerkin finite element scheme for the Cahn-Hilliard equation. Math. Comp. 88, 211–235 (2019)
36. Wang, X., Meng, X., Zhang, S., Zhou, H.: A modified weak Galerkin finite element method for the linear elasticity problem in mixed form. J. Comput. Appl. Math. 420, Paper No. 114743, 19 (2023)
37. Wang, X., Ye, X., Zhang, S., Zhu, P.: A weak Galerkin least squares finite element method of Cauchy problem for Poisson equation. J. Comput. Appl. Math. 401, Paper No. 113767, 9 (2022)
38. Wang, X., Zhai, Q., Zhang, R., Zhang, S.: The weak Galerkin finite element method for solving the time-dependent integro-differential equations. Adv. Appl. Math. Mech. 12(1), 164–188 (2020)
39. Xie, Y., Zhong, L.: Convergence of adaptive weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 86, 17 (2021)
40. Ye, X., Zhang, S.: A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes. SIAM J. Numer. Anal. 58, 2572–2588 (2020)
41. Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method. Int. J. Numer. Anal. Model. 17, 110–117 (2020)
42. Ye, X., Zhang, S.: A conforming discontinuous Galerkin finite element method: Part II. Int. J. Numer. Anal. Model. 17, 281–296 (2020)
43. Ye, X., Zhang, S.: A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh. Electron. Res. Arch. 29(6), 3609–3627 (2021)
44. Ye, X., Zhang, S.: A numerical scheme with divergence free H-div triangular finite element for the Stokes equations. Appl. Numer. Math. 167, 211–217 (2021)
45. Ye, X., Zhang, S.: A weak Galerkin finite element method for p-Laplacian problem. East Asian J. Appl. Math. 11(2), 219–233 (2021)
46. Ye, X., Zhang, S.: Order two superconvergence of the CDG method for the Stokes equations on triangle/tetrahedron. J. Appl. Analys. Comput 12(6), 2578–2592 (2022)
47. Ye, X., Zhang, S.: A C⁰-conforming DG finite element method for biharmonic equations on triangle/tetrahedron. J. Numer. Math. 30(3), 163–172 (2022)
48. Ye, X., Zhang, S.: Achieving superconvergence by one-dimensional discontinuous finite elements: the CDG method. East Asian J. Appl. Math. 12(4), 781–790 (2022)
49. Ye, X., Zhang, S.: A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes. Appl. Numer. Math. 178, 155–165 (2022)
50. Ye, X., Zhang, S.: Achieving superconvergence by one-dimensional discontinuous finite elements: weak Galerkin method. East Asian J. Appl. Math. 12(3), 590–598 (2022)
51. Ye, X., Zhang, S.: Achieving superconvergence by one-dimensional discontinuous finite elements: the CDG method. East Asian J. Appl. Math. 12(4), 781–790 (2022)
52. Ye, X., Zhang, S.: Order two superconvergence of the CDG finite elements on triangular and tetrahedral meshes. CSIAM Trans. Appl. Math. 4(2), 256–274 (2023)
53. Ye, X., Zhang, S.: Constructing order two superconvergent WG finite elements on rectangular meshes. Numer. Math. Theory Methods Appl. 16(1), 230–241 (2023)
54. Ye, X., Zhang, S.: Four-order superconvergent weak Galerkin methods for the biharmonic equation on triangular meshes. Communications on Applied Mathematics and Computation 5, 1323–1338 (2023). https:// doi. org/ 10. 1007/ s42967- 022- 00201-5
55. Ye, X., Zhang, S., Zhang, Z.: A new P₁ weak Galerkin method for the biharmonic equation. J. Comput. Appl. Math. 364, 12337 (2020)
56. Ye, X., Zhang, S., Zhang, Z.: A locking-free weak Galerkin finite element method for ReissnerMindlin plate on polygonal meshes. Comput. Math. Appl. 80(5), 906–916 (2020)
57. Ye, X., Zhang, S., Zhu, P.: A weak Galerkin finite element method for nonlinear conservation laws. Electron. Res. Arch. 29(1), 1897–1923 (2021)
58. Zhang, H., Zou, Y., Chai, S., Yue, H.: Weak Galerkin method with (r, r-1, r-1)-order finite elements for second order parabolic equations. Appl. Math. Comp. 275, 24–40 (2016)
59. Zhang, J., Zhang, K., Li, J., Wang, X.: A weak Galerkin finite element method for the NavierStokes equations. Commun. Comput. Phys 23, 706–746 (2018)
60. Zhang, T., Lin, T.: A posteriori error estimate for a modified weak Galerkin method solving elliptic problems. Numer. Methods Partial Differential Equations 33, 381–398 (2017)
61. Zhang, T., Zhang, S.: The weak Galerkin finite element method for the symmetric hyperbolic systems. J. Comput. Appl. Math. 365, 112375 (2020)
62. Zhang, T., Zhang, S.: The weak Galerkin finite element method for the transport-reaction equation. J. Comput. Phys. 410, 109399 (2020)
63. Zhang, T., Zhang, S.: An explicit weak Galerkin method for solving the first order hyperbolic systems. J. Comput. Appl. Math. 412, Paper No. 114311 (2022)
64. Zhou, S., Gao, F., Li, B., Sun, Z.: Weak Galerkin finite element method with second-order accuracy in time for parabolic problems. Appl. Math. Lett. 90, 118–123 (2019)
65. Zhu, A., Xu, T., Xu, Q.: Weak Galerkin finite element methods for linear parabolic integro-differential equations. Numer. Methods Partial Differential Equations 32, 1357–1377 (2016)

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