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A systematic study on weak Galerkin finite element methods for second order elliptic problems. (English) Zbl 1398.65311

Summary: This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form \(P_k(T)\times P_j(\partial T)\| P_\ell (T)^2\), where \(k\geq 1\) is the degree of polynomials in the interior of the element \(T\), \(j\geq 0\) is the degree of polynomials on the boundary of \(T\), and \(\ell \geq 0\) is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Discontinuous Galerkin methods for elliptic problems. Discontinuous Galerkin methods (Newport, RI, 1999), 89-101. Lecture notes in computational science and engineering. Springer, Berlin (2000) · Zbl 0948.65127
[2] Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749-1779 (2001) · Zbl 1008.65080 · doi:10.1137/S0036142901384162
[3] Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23, 199-214 (2013) · Zbl 1416.65433 · doi:10.1142/S0218202512500492
[4] Beirão da Veiga, L., Manzini, G.: A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34, 759-781 (2014) · Zbl 1293.65146 · doi:10.1093/imanum/drt018
[5] Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455-462 (2013) · Zbl 1297.74049 · doi:10.1016/j.cma.2012.09.012
[6] Chen, L., Wang, J., Ye, X.: A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 59, 496-511 (2014) · Zbl 1307.65153 · doi:10.1007/s10915-013-9771-3
[7] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) · Zbl 0383.65058
[8] Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319-1365 (2009) · Zbl 1205.65312 · doi:10.1137/070706616
[9] Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1-21 (2015) · Zbl 1423.74876 · doi:10.1016/j.cma.2014.09.009
[10] Gao, F., Mu, L.: On \[L^2\] L2 error estimate for weak Galerkin finite element methods for parabolic problems. J. Comput. Math. 32, 195-204 (2014) · Zbl 1313.65246 · doi:10.4208/jcm.1401-m4385
[11] Huang, W., Wang, Y.: Discrete maximum principle for the weak Galerkin method for anisotropic diffusion problems. Commun. Comput. Phys. 18, 65-90 (2015) · Zbl 1388.65151 · doi:10.4208/cicp.180914.121214a
[12] Karakashian, O.A., Pascal, F.: A posterior error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374-2399 (2003) · Zbl 1058.65120 · doi:10.1137/S0036142902405217
[13] Lehrenfeld, C.: Hybrid Discontinuous Galerkin methods for solving incompressible flow problems, Diploma Thesis, MathCCES/IGPM, RWTH Aachen. (2010) · Zbl 1008.65080
[14] Li, B., Xie, X.: A two-level algorithm for the weak Galerkin discretization of diffusion problems. J. Comput. Appl. Math. 287, 179-195 (2015) · Zbl 1320.65177 · doi:10.1016/j.cam.2015.03.043
[15] Li, J., Wang, X., Zhang, K.: Multi-level Monte Carlo weak Galerkin method for elliptic equations with stochastic jump coefficients. Appl. Math. Comput. 275, 181-194 (2016) · Zbl 1410.65454 · doi:10.1016/j.amc.2015.11.064
[16] Li, Q., Wang, J.: Weak Galerkin finite element methods for parabolic equations. Numer. Methods Partial Differ. Equ. 29, 2004-2024 (2013) · Zbl 1307.65133
[17] Mu, L., Wang, J., Wang, Y., Ye, X.: A computational study of the weak Galerkin method for second-order elliptic equations. Numer. Algorithms 63, 753-777 (2013) · Zbl 1271.65140 · doi:10.1007/s11075-012-9651-1
[18] Mu, L., Wang, J., Ye, X.: A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J. Comput. Phys. 273, 327-342 (2014) · Zbl 1351.76072 · doi:10.1016/j.jcp.2014.04.017
[19] Mu, L., Wang, J., Ye, X.: A new weak Galerkin finite element method for the Helmholtz equation. IMA J. Numer. Anal. 35, 1228-1255 (2015) · Zbl 1323.65116 · doi:10.1093/imanum/dru026
[20] Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Model. 12, 31-53 (2015) · Zbl 1332.65172
[21] Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comput. Appl. Math. 285, 45-58 (2015) · Zbl 1315.65099 · doi:10.1016/j.cam.2015.02.001
[22] Mu, L., Wang, J., Ye, X., Zhang, S.: A \[C^0\] C0-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 59, 473-495 (2014) · Zbl 1305.65233 · doi:10.1007/s10915-013-9770-4
[23] Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65, 363-386 (2015) · Zbl 1327.65220 · doi:10.1007/s10915-014-9964-4
[24] Mu, L., Wang, J., Ye, X., Zhao, S.: A numerical study on the weak Galerkin method for the Helmholtz equation. Commun. Comput. Phys. 15, 1461-1479 (2014) · Zbl 1388.65156 · doi:10.4208/cicp.251112.211013a
[25] Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math. Appl. 68, 2314-2330 (2014) · Zbl 1361.35058 · doi:10.1016/j.camwa.2014.03.021
[26] Wang, C., Wang, J.: A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form. Math. Comp. (2017). doi:10.1090/mcom/3220
[27] Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103-115 (2013) · Zbl 1261.65121 · doi:10.1016/j.cam.2012.10.003
[28] Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comp. 83, 2101-2126 (2014) · Zbl 1308.65202 · doi:10.1090/S0025-5718-2014-02852-4
[29] Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. 42, 155-174 (2016) · Zbl 1382.76178 · doi:10.1007/s10444-015-9415-2
[30] Wang, R., Wang, X., Zhai, Q., Zhang, R.: A weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. 302, 171-185 (2016) · Zbl 1337.65162 · doi:10.1016/j.cam.2016.01.025
[31] Wang, X., Zhai, Q., Zhang, R.: The weak Galerkin method for solving the incompressible Brinkman flow. J. Comput. Appl. Math. 307, 13-24 (2016) · Zbl 1338.76069 · doi:10.1016/j.cam.2016.04.031
[32] Zhai, Q., Zhang, R., Mu, L.: A new weak Galerkin finite element scheme for the Brinkman model. Commun. Comput. Phys. 19, 1409-1434 (2016) · Zbl 1373.76108 · doi:10.4208/cicp.scpde14.44s
[33] Zhang, R., Zhai, Q.: A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J. Sci. Comput. 64, 559-585 (2015) · Zbl 1331.65163 · doi:10.1007/s10915-014-9945-7
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