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A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems

  • * Corresponding author: Lunji Song

    * Corresponding author: Lunji Song 
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  • 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.

    Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 35j50.

    Citation:

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  • Figure 1.  Initial mesh (Left) and the OPWG ($ \beta_{0} = 3 $) solution on the finest mesh (Right) for Example 2

    Figure 2.  Convergence rates of the OPWG ($ \beta_{0} = 3 $) solutions against degree of freedoms for different values of $ \alpha $ in Example 2. (Left) $ \alpha = 0.5 $; (Right) $ \alpha = 0.25 $

    Table 1.  WG method with element $ (\mathbb{P}_k, \mathbb{P}_{k}, \mathbb{P}_{k-1}^2) $ for Example 1

    $ h $ $ k=1 $ $ k=2 $
    $ ||| e_{h}||| $ $ \|e_{0}\| $ $ ||| e_{h}||| $ $ \|e_{0}\| $
    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
     | Show Table
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    Table 2.  OPWG with $ (\mathbb{P}_1, \mathbb{P}_1, \mathbb{P}_{0}^{2}) $ and $ \beta_0 = 1, 2, 3, 4 $ for Example 1

    $ h $ $ ||| e_h ||| $ $ \|e_0\| $ $ ||| e_h ||| $ $ \|e_0\| $
    $ \beta_0=1 $ $ \beta_0=2 $
    $ 1/8 $ 1.3258e+00 7.4931e-02 1.0550e+00 5.9839e-02
    $ 1/16 $ 1.0306e+00 3.4203e-02 5.6097e-01 1.5700e-02
    $ 1/32 $ 9.4663e-01 2.5074e-02 3.1098e-01 4.3112e-03
    $ 1/64 $ 9.2663e-01 2.3053e-02 1.8215e-01 1.2815e-03
    Rate. 0.0308 0.1212 0.7717 1.7503
    $ \beta_0=3 $ $ \beta_0=4 $
    $ 1/8 $ 1.0019e+00 5.7508e-02 9.9304e-01 5.7135e-02
    $ 1/16 $ 5.0137e-01 1.4381e-02 4.9628e-01 1.4278e-02
    $ 1/32 $ 2.5075e-01 3.5954e-03 2.4804e-01 3.5682e-03
    $ 1/64 $ 1.2539e-01 8.9887e-04 1.2399e-01 8.9191e-04
    Rate. 0.9998 2. 1.0003 2.0002
     | Show Table
    DownLoad: CSV

    Table 3.  OPWG with $ (\mathbb{P}_2, \mathbb{P}_2, \mathbb{P}_{1}^{2}) $ and $ \beta_0 = 2, 3, 4, 5 $ for Example 1

    $ h $ $ ||| e_h ||| $ $ \|e_0\| $ $ ||| e_h ||| $ $ \|e_0\| $
    $ \beta_0=2 $ $ \beta_0=3 $
    $ 1/8 $ 1.7917e+00 1.0256e-01 8.4595e-01 2.0487e-02
    $ 1/16 $ 1.4060e+00 5.7784e-02 4.4264e-01 5.3334e-03
    $ 1/32 $ 1.0580e+00 3.0997e-02 2.2464e-01 1.3530e-03
    $ 1/64 $ 7.7468e-01 1.6118e-02 1.1291e-01 3.4023e-04
    Rate. 0.4497 0.9435 0.9924 1.9916
    $ \beta_0=4 $ $ \beta_0=5 $
    $ 1/8 $ 3.6348e-01 4.6229e-03 1.6594e-01 3.2492e-03
    $ 1/16 $ 1.2933e-01 5.9131e-04 4.1839e-02 4.0604e-04
    $ 1/32 $ 4.5730e-02 7.4938e-05 1.0494e-02 5.0757e-05
    $ 1/64 $ 1.6162e-02 9.4394e-06 2.6274e-03 6.3443e-06
    Rate. 1.5005 2.9889 1.9979 3.0001
     | Show Table
    DownLoad: CSV

    Table 4.  Comparison of condition number with optimal penalty parameters

    $ h $ Without preconditioning Block-diagonal preconditioning
    $ k=1, \beta_{0}=3 $ $ k=2, \beta_{0}=5 $ $ k=1, \beta_{0}=3 $ $ k=2, \beta_{0}=5 $
    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
     | Show Table
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    Table 5.  Errors and condition numbers for Example 1 with $ (\mathbb{P}_1, \mathbb{P}_0, \mathbb{P}_{0}^{2}) $ and $ \beta_0 = 3 $

    $ h $ $ ||| e_h ||| $ $ \|e_0\| $ Cond. Pre. Cond.
    $ 1/8 $ 1.5226e+00 8.7459e-02 6.6551e+03 2.1055e+03
    $ 1/16 $ 7.7405e-01 2.2035e-02 8.1739e+04 7.9132e+03
    $ 1/32 $ 3.8921e-01 5.5253e-03 1.2109e+06 3.1176e+04
    $ 1/64 $ 1.9499e-01 1.3832e-03 1.9007e+07 1.2425e+05
    Rate. 0.9971 1.9980 -3.9724 -1.9947
     | Show Table
    DownLoad: CSV

    Table 6.  Errors and condition numbers for Example 1 with $ (\mathbb{P}_2, \mathbb{P}_1, \mathbb{P}_{1}^{2}) $ and $ \beta_0 = 5 $

    $ h $ $ ||| e_h ||| $ $ \|e_0\| $ Cond. Pre. Cond.
    $ 1/8 $ 1.6622e-01 3.3354e-03 3.5419e+05 1.0229e+05
    $ 1/16 $ 4.1912e-02 4.1731e-04 1.9810e+07 4.2176e+05
    $ 1/32 $ 1.0513e-02 5.2184e-05 1.2227e+09 1.7166e+06
    $ 1/64 $ 2.6321e-03 6.5231e-06 7.7586e+10 6.9742e+06
    Rate. 1.9979 3.0000 -5.9877 -2.0225
     | Show Table
    DownLoad: CSV

    Table 7.  OPWG with $ (\mathbb{P}_1, \mathbb{P}_1, \mathbb{P}_{0}^{2}) $ and optimal penalty parameter for Example 2

    dof. $ \alpha=0.5 $ $ \alpha=0.25 $ Condition Number
    $ ||| e_h ||| $ $ \|e_{0}\| $ $ ||| e_h ||| $ $ \|e_{0}\| $ 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
     | Show Table
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