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A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems
1. | School of Mathematics and Statistics, and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China |
2. | Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA |
In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements $ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{d}) $, or $ (\mathbb{P}_{k}, \mathbb{P}_{k-1}, [\mathbb{P}_{k-1}]^{d}) $, with dimensions of space $ d = 2, \; 3 $. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shape-regularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are $ O(h^{-\beta_{0}(d-1)-1}) $, $ \beta_{0} $ being the penalty exponent. Therefore we introduce a new mini-block diagonal preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of $ O(h^{-2}) $. Optimal error estimates in a discrete $ H^1 $-norm and $ L^2 $-norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.
References:
[1] |
D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,
Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.
doi: 10.1137/S0036142901384162. |
[2] |
F. Brezzi, J. Douglas Jr. and L. D. Marini,
Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.
doi: 10.1007/BF01389710. |
[3] |
B. Li and X. Xie,
A two-level algorithm for the weak Galerkin discretization of diffusion problems, J. Comput. Appl. Math., 287 (2015), 179-195.
doi: 10.1016/j.cam.2015.03.043. |
[4] |
K. Liu, L. Song and S. Zhou,
An over-penalized weak Galerkin method for second-order elliptic problems, J. Comput. Math., 36 (2018), 866-880.
doi: 10.4208/jcm.1705-m2016-0744. |
[5] |
L. Mu, J. Wang, Y. Wang and X. Ye,
A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algor., 63 (2012), 753-777.
doi: 10.1007/s11075-012-9651-1. |
[6] |
L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao,
Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250 (2013), 106-125.
doi: 10.1016/j.jcp.2013.04.042. |
[7] |
L. Mu, J. Wang and X. Ye,
A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35 (2015), 1228-1255.
doi: 10.1093/imanum/dru026. |
[8] |
L. Mu, J. Wang and X. Ye,
A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285 (2015), 45-58.
doi: 10.1016/j.cam.2015.02.001. |
[9] |
L. Mu, J. Wang and X. Ye,
Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.
|
[10] |
L. Mu, J. Wang, X. Ye and S. Zhang,
A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386.
doi: 10.1007/s10915-014-9964-4. |
[11] |
A. Quarteroni and V. Alberto, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer, Berlin, Heidelberg, 1994.
doi: 10.1007/978-3-540-85268-1. |
[12] |
P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Galligani I., Magenes E. (eds) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol 606, Springer, Berlin, Heidelberg.
doi: https://doi.org/10.1007/BFb0064470. |
[13] |
B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008.
doi: doi.org/10.1137/1.9780898717440. |
[14] |
L. Song, K. Liu and S. Zhao,
A weak Galerkin method with an over-relaxed stabilization for low regularity elliptic problems, J. Sci. Comput., 71 (2017), 195-218.
doi: 10.1007/s10915-016-0296-4. |
[15] |
L. Song, S. Zhao and K. Liu,
A relaxed weak Galerkin method for elliptic interface problems with low regularity, Appl. Numer. Math., 128 (2018), 65-80.
doi: 10.1016/j.apnum.2018.01.021. |
[16] |
C. Wang and J. Wang,
An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68 (2014), 2314-2330.
doi: 10.1016/j.camwa.2014.03.021. |
[17] |
J. Wang and X. Ye,
A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.
doi: 10.1016/j.cam.2012.10.003. |
[18] |
J. Wang and X. Ye,
A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126.
doi: 10.1090/S0025-5718-2014-02852-4. |
[19] |
J. Wang and X. Ye,
A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174.
doi: 10.1007/s10444-016-9471-2. |
[20] |
Q. Zhai, X. Ye, R. Wang and R. Zhang,
A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems, Comput. Math. Appl., 74 (2017), 2243-2252.
doi: 10.1016/j.camwa.2017.07.009. |
show all references
References:
[1] |
D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,
Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.
doi: 10.1137/S0036142901384162. |
[2] |
F. Brezzi, J. Douglas Jr. and L. D. Marini,
Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.
doi: 10.1007/BF01389710. |
[3] |
B. Li and X. Xie,
A two-level algorithm for the weak Galerkin discretization of diffusion problems, J. Comput. Appl. Math., 287 (2015), 179-195.
doi: 10.1016/j.cam.2015.03.043. |
[4] |
K. Liu, L. Song and S. Zhou,
An over-penalized weak Galerkin method for second-order elliptic problems, J. Comput. Math., 36 (2018), 866-880.
doi: 10.4208/jcm.1705-m2016-0744. |
[5] |
L. Mu, J. Wang, Y. Wang and X. Ye,
A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algor., 63 (2012), 753-777.
doi: 10.1007/s11075-012-9651-1. |
[6] |
L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao,
Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250 (2013), 106-125.
doi: 10.1016/j.jcp.2013.04.042. |
[7] |
L. Mu, J. Wang and X. Ye,
A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35 (2015), 1228-1255.
doi: 10.1093/imanum/dru026. |
[8] |
L. Mu, J. Wang and X. Ye,
A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285 (2015), 45-58.
doi: 10.1016/j.cam.2015.02.001. |
[9] |
L. Mu, J. Wang and X. Ye,
Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.
|
[10] |
L. Mu, J. Wang, X. Ye and S. Zhang,
A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386.
doi: 10.1007/s10915-014-9964-4. |
[11] |
A. Quarteroni and V. Alberto, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer, Berlin, Heidelberg, 1994.
doi: 10.1007/978-3-540-85268-1. |
[12] |
P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Galligani I., Magenes E. (eds) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol 606, Springer, Berlin, Heidelberg.
doi: https://doi.org/10.1007/BFb0064470. |
[13] |
B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008.
doi: doi.org/10.1137/1.9780898717440. |
[14] |
L. Song, K. Liu and S. Zhao,
A weak Galerkin method with an over-relaxed stabilization for low regularity elliptic problems, J. Sci. Comput., 71 (2017), 195-218.
doi: 10.1007/s10915-016-0296-4. |
[15] |
L. Song, S. Zhao and K. Liu,
A relaxed weak Galerkin method for elliptic interface problems with low regularity, Appl. Numer. Math., 128 (2018), 65-80.
doi: 10.1016/j.apnum.2018.01.021. |
[16] |
C. Wang and J. Wang,
An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68 (2014), 2314-2330.
doi: 10.1016/j.camwa.2014.03.021. |
[17] |
J. Wang and X. Ye,
A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.
doi: 10.1016/j.cam.2012.10.003. |
[18] |
J. Wang and X. Ye,
A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126.
doi: 10.1090/S0025-5718-2014-02852-4. |
[19] |
J. Wang and X. Ye,
A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174.
doi: 10.1007/s10444-016-9471-2. |
[20] |
Q. Zhai, X. Ye, R. Wang and R. Zhang,
A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems, Comput. Math. Appl., 74 (2017), 2243-2252.
doi: 10.1016/j.camwa.2017.07.009. |


1/8 | 9.9173e-01 | 5.3131e-02 | 7.8508e-02 | 3.2464e-03 |
1/16 | 4.9588e-01 | 1.3272e-02 | 1.9677e-02 | 4.0611e-04 |
1/32 | 2.4793e-01 | 3.3169e-03 | 4.9226e-03 | 5.0761e-05 |
1/64 | 1.2396e-01 | 8.2914e-04 | 1.2309e-03 | 6.3445e-06 |
Rate. | 1.0001 | 2.0001 | 1.9997 | 3.0001 |
1/8 | 9.9173e-01 | 5.3131e-02 | 7.8508e-02 | 3.2464e-03 |
1/16 | 4.9588e-01 | 1.3272e-02 | 1.9677e-02 | 4.0611e-04 |
1/32 | 2.4793e-01 | 3.3169e-03 | 4.9226e-03 | 5.0761e-05 |
1/64 | 1.2396e-01 | 8.2914e-04 | 1.2309e-03 | 6.3445e-06 |
Rate. | 1.0001 | 2.0001 | 1.9997 | 3.0001 |
|
1.3258e+00 | 7.4931e-02 | 1.0550e+00 | 5.9839e-02 |
1.0306e+00 | 3.4203e-02 | 5.6097e-01 | 1.5700e-02 | |
9.4663e-01 | 2.5074e-02 | 3.1098e-01 | 4.3112e-03 | |
9.2663e-01 | 2.3053e-02 | 1.8215e-01 | 1.2815e-03 | |
Rate. | 0.0308 | 0.1212 | 0.7717 | 1.7503 |
|
1.0019e+00 | 5.7508e-02 | 9.9304e-01 | 5.7135e-02 |
5.0137e-01 | 1.4381e-02 | 4.9628e-01 | 1.4278e-02 | |
2.5075e-01 | 3.5954e-03 | 2.4804e-01 | 3.5682e-03 | |
1.2539e-01 | 8.9887e-04 | 1.2399e-01 | 8.9191e-04 | |
Rate. | 0.9998 | 2. | 1.0003 | 2.0002 |
|
1.3258e+00 | 7.4931e-02 | 1.0550e+00 | 5.9839e-02 |
1.0306e+00 | 3.4203e-02 | 5.6097e-01 | 1.5700e-02 | |
9.4663e-01 | 2.5074e-02 | 3.1098e-01 | 4.3112e-03 | |
9.2663e-01 | 2.3053e-02 | 1.8215e-01 | 1.2815e-03 | |
Rate. | 0.0308 | 0.1212 | 0.7717 | 1.7503 |
|
1.0019e+00 | 5.7508e-02 | 9.9304e-01 | 5.7135e-02 |
5.0137e-01 | 1.4381e-02 | 4.9628e-01 | 1.4278e-02 | |
2.5075e-01 | 3.5954e-03 | 2.4804e-01 | 3.5682e-03 | |
1.2539e-01 | 8.9887e-04 | 1.2399e-01 | 8.9191e-04 | |
Rate. | 0.9998 | 2. | 1.0003 | 2.0002 |
|
1.7917e+00 | 1.0256e-01 | 8.4595e-01 | 2.0487e-02 |
1.4060e+00 | 5.7784e-02 | 4.4264e-01 | 5.3334e-03 | |
1.0580e+00 | 3.0997e-02 | 2.2464e-01 | 1.3530e-03 | |
7.7468e-01 | 1.6118e-02 | 1.1291e-01 | 3.4023e-04 | |
Rate. | 0.4497 | 0.9435 | 0.9924 | 1.9916 |
|
3.6348e-01 | 4.6229e-03 | 1.6594e-01 | 3.2492e-03 |
1.2933e-01 | 5.9131e-04 | 4.1839e-02 | 4.0604e-04 | |
4.5730e-02 | 7.4938e-05 | 1.0494e-02 | 5.0757e-05 | |
1.6162e-02 | 9.4394e-06 | 2.6274e-03 | 6.3443e-06 | |
Rate. | 1.5005 | 2.9889 | 1.9979 | 3.0001 |
|
1.7917e+00 | 1.0256e-01 | 8.4595e-01 | 2.0487e-02 |
1.4060e+00 | 5.7784e-02 | 4.4264e-01 | 5.3334e-03 | |
1.0580e+00 | 3.0997e-02 | 2.2464e-01 | 1.3530e-03 | |
7.7468e-01 | 1.6118e-02 | 1.1291e-01 | 3.4023e-04 | |
Rate. | 0.4497 | 0.9435 | 0.9924 | 1.9916 |
|
3.6348e-01 | 4.6229e-03 | 1.6594e-01 | 3.2492e-03 |
1.2933e-01 | 5.9131e-04 | 4.1839e-02 | 4.0604e-04 | |
4.5730e-02 | 7.4938e-05 | 1.0494e-02 | 5.0757e-05 | |
1.6162e-02 | 9.4394e-06 | 2.6274e-03 | 6.3443e-06 | |
Rate. | 1.5005 | 2.9889 | 1.9979 | 3.0001 |
Without preconditioning | Block-diagonal preconditioning | |||
1/4 | 1.8272e+04 | 2.1075e+04 | 2.6265e+02 | 4.3079e+04 |
1/8 | 1.6957e+05 | 6.9533e+05 | 1.0135e+03 | 1.9257e+05 |
1/16 | 2.1585e+06 | 3.4759e+07 | 4.0481e+03 | 8.0866e+05 |
1/32 | 3.2471e+07 | 2.0653e+09 | 1.6228e+04 | 3.3427e+06 |
1/64 | 5.1218e+08 | 1.2974e+11 | 6.4995e+04 | 1.3570e+07 |
Order | -3.9794 | -5.9731 | -2.0018 | -2.0213 |
Without preconditioning | Block-diagonal preconditioning | |||
1/4 | 1.8272e+04 | 2.1075e+04 | 2.6265e+02 | 4.3079e+04 |
1/8 | 1.6957e+05 | 6.9533e+05 | 1.0135e+03 | 1.9257e+05 |
1/16 | 2.1585e+06 | 3.4759e+07 | 4.0481e+03 | 8.0866e+05 |
1/32 | 3.2471e+07 | 2.0653e+09 | 1.6228e+04 | 3.3427e+06 |
1/64 | 5.1218e+08 | 1.2974e+11 | 6.4995e+04 | 1.3570e+07 |
Order | -3.9794 | -5.9731 | -2.0018 | -2.0213 |
Cond. | Pre. Cond. | |||
|
1.5226e+00 | 8.7459e-02 | 6.6551e+03 | 2.1055e+03 |
7.7405e-01 | 2.2035e-02 | 8.1739e+04 | 7.9132e+03 | |
3.8921e-01 | 5.5253e-03 | 1.2109e+06 | 3.1176e+04 | |
1.9499e-01 | 1.3832e-03 | 1.9007e+07 | 1.2425e+05 | |
Rate. | 0.9971 | 1.9980 | -3.9724 | -1.9947 |
Cond. | Pre. Cond. | |||
|
1.5226e+00 | 8.7459e-02 | 6.6551e+03 | 2.1055e+03 |
7.7405e-01 | 2.2035e-02 | 8.1739e+04 | 7.9132e+03 | |
3.8921e-01 | 5.5253e-03 | 1.2109e+06 | 3.1176e+04 | |
1.9499e-01 | 1.3832e-03 | 1.9007e+07 | 1.2425e+05 | |
Rate. | 0.9971 | 1.9980 | -3.9724 | -1.9947 |
Cond. | Pre. Cond. | |||
|
1.6622e-01 | 3.3354e-03 | 3.5419e+05 | 1.0229e+05 |
4.1912e-02 | 4.1731e-04 | 1.9810e+07 | 4.2176e+05 | |
1.0513e-02 | 5.2184e-05 | 1.2227e+09 | 1.7166e+06 | |
2.6321e-03 | 6.5231e-06 | 7.7586e+10 | 6.9742e+06 | |
Rate. | 1.9979 | 3.0000 | -5.9877 | -2.0225 |
Cond. | Pre. Cond. | |||
|
1.6622e-01 | 3.3354e-03 | 3.5419e+05 | 1.0229e+05 |
4.1912e-02 | 4.1731e-04 | 1.9810e+07 | 4.2176e+05 | |
1.0513e-02 | 5.2184e-05 | 1.2227e+09 | 1.7166e+06 | |
2.6321e-03 | 6.5231e-06 | 7.7586e+10 | 6.9742e+06 | |
Rate. | 1.9979 | 3.0000 | -5.9877 | -2.0225 |
dof. | Condition Number | |||||
Cond. | Pre. Cond. | |||||
2.0880e+3 | 4.2303e-1 | 7.0060e-2 | 9.6614e-1 | 7.3906e-2 | 1.7335e+6 | 1.1357e+3 |
8.3520e+3 | 2.7461e-1 | 1.8365e-2 | 7.8292e-1 | 2.2116e-2 | 2.4882e+7 | 4.5578e+3 |
3.3408e+4 | 1.8503e-1 | 4.7471e-3 | 6.4655e-1 | 7.3419e-3 | 3.8905e+8 | 1.8270e+4 |
1.3363e+5 | 1.3025e-1 | 1.2371e-3 | 5.5838e-1 | 2.8274e-3 | 6.1854e+9 | 7.3140e+4 |
5.3453e+5 | 9.2125e-2 | 3.2954e-4 | 4.6994e-1 | 1.1394e-3 | 9.8828e+10 | 2.9267e+5 |
dof. | Condition Number | |||||
Cond. | Pre. Cond. | |||||
2.0880e+3 | 4.2303e-1 | 7.0060e-2 | 9.6614e-1 | 7.3906e-2 | 1.7335e+6 | 1.1357e+3 |
8.3520e+3 | 2.7461e-1 | 1.8365e-2 | 7.8292e-1 | 2.2116e-2 | 2.4882e+7 | 4.5578e+3 |
3.3408e+4 | 1.8503e-1 | 4.7471e-3 | 6.4655e-1 | 7.3419e-3 | 3.8905e+8 | 1.8270e+4 |
1.3363e+5 | 1.3025e-1 | 1.2371e-3 | 5.5838e-1 | 2.8274e-3 | 6.1854e+9 | 7.3140e+4 |
5.3453e+5 | 9.2125e-2 | 3.2954e-4 | 4.6994e-1 | 1.1394e-3 | 9.8828e+10 | 2.9267e+5 |
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