    May  2021, 26(5): 2581-2598. doi: 10.3934/dcdsb.2020196

## A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems

 School of Mathematics and Statistics and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China

* Corresponding author: Lunji Song

Received  November 2019 Revised  March 2020 Published  June 2020

Fund Project: Song's research was supported by the Natural Science Foundation of Gansu Province, China (Grant 18JR3RA290)

We introduce an over-penalized weak Galerkin method for elliptic interface problems with non-homogeneous boundary conditions and discontinuous coefficients, where the method combines a weak Galerkin stabilizer with interior penalty terms. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements $(\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{2})$. As an advantage of the method, elliptic interface problems with low regularity are approximated well. The over-penalized weak Galerkin method is based on weak functions whose edge part is double-valued on each interior edge sharing by two neighboring elements. Jumps between the edge parts are naturally used to define penalty terms. The over-penalized weak Galerkin method allows to use arbitrary meshes, even for low regularity solutions. These features make the new method more flexible and efficient for solving interface equations. Furthermore, a priori error estimates in energy and $L^{2}$ norms are derived rigorously, and numerical results confirm the effectiveness of the method.

Citation: Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
##### References:
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show all references

##### References:
  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.  Google Scholar  E. Burman and P. Hansbo, Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems, IMA J. Numer. Anal., 30 (2010), 870-885.  doi: 10.1093/imanum/drn081.  Google Scholar  Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.  doi: 10.1007/s002110050336.  Google Scholar  G. R. Hadley, High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners, J. Lightwave Technol., 20 (2002), 1219-1231.  doi: 10.1109/JLT.2002.800371. Google Scholar  S. Hou, Z. Lin, L. Wang and W. Wang, A numerical method for solving elasticity equations with interfaces, Commun. Comput. Phys., 12 (2012), 595-612.  doi: 10.4208/cicp.160910.130711s.  Google Scholar  S. Hou, W. Wang and L. Wang, Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces, J. Comput. Phys., 229 (2010), 7162-7179.  doi: 10.1016/j.jcp.2010.06.005.  Google Scholar  T. Y. Hou, Z. Li and S. Osher, Hybrid method for moving interface problems with application to the Hele-Shaw flow, J. Comput. Phys., 134 (1997), 236-252.  doi: 10.1006/jcph.1997.5689.  Google Scholar  A. T. Layton, Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces, Comput. & Fluids, 38 (2009), 266-272.  doi: 10.1016/j.compfluid.2008.02.003.  Google Scholar  R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singular perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56 (2018), 1482-1497.  doi: 10.1137/17M1152528.  Google Scholar  L. Mu, Weak Galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes, J. Comput. Appl. Math., 361 (2019), 413-425.  doi: 10.1016/j.cam.2019.04.026.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  L. Mu, J. Wang, X. Ye and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys., 325 (2016), 157-173.  doi: 10.1016/j.jcp.2016.08.024.  Google Scholar  W. Qi and L. Song, Weak Galerkin method with implicit $\theta$-schemes for second-order parabolic problems, Appl. Math. Comput., 366 (2020), 11pp. doi: 10.1016/j.amc.2019.124731.  Google Scholar  P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., 606, Springer, Berlin, 1977. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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/s10444-015-9415-2.  Google Scholar  J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.  Google Scholar  J. Wang and X. Ye, The basics of weak Galerkin finite element methods, preprint, arXiv: 1901.10035. Google Scholar  Y. C. Zhou and G. W. Wei, On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, J. Comput. Phys., 219 (2006), 228-246.  doi: 10.1016/j.jcp.2006.03.027.  Google Scholar Example 1: piecewise linear elements. Left: numerical solution, Right: error Example 1: piecewise quadratic elements. Left: numerical solution, Right: error Example 2: piecewise linear elements, with $\beta_0 = 3$, $\mbox{Level} = 5$. Left: numerical solution, Right: error Example 2: piecewise quadratic elements, with $\beta_0 = 5$, $\mbox{Level} = 5$. Left: numerical solution, Right: error Example 3: piecewise linear elements, with $\beta_0 = 3$, $\mbox{Level} = 5$. Left: numerical solution, Right: error Example 3: piecewise quadratic elements, with $\beta_0 = 5$, $\mbox{Level} = 5$. Left: numerical solution, Right: error Example 4: piecewise linear elements. Left: the initial mesh, Right: numerical solution with $\beta_0 = 3$, $\mbox{Level} = 5$
Example 1 - piecewise linear elements (k = 1)
 Mesh $\beta_{0}=1$ $\beta_{0}=2$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.1511e+0 2.8316e+0 3.1452e+0 1.1782e+0 Level 2 1.6608e+0 0.9240 3.4495e+0 -0.2848 1.6225e+0 0.9549 6.7356e-1 0.8067 Level 3 9.6287e-1 0.7865 3.7937e+0 -0.1372 8.5141e-1 0.9303 3.5846e-1 0.9100 Level 4 6.7713e-1 0.5079 3.9819e+0 -0.0699 4.5318e-1 0.9098 1.8459e-1 0.9575 Level 5 5.8043e-1 0.2223 4.0839e+0 -0.0365 2.4081e-1 0.9122 9.3619e-2 0.9795
 Mesh $\beta_{0}=1$ $\beta_{0}=2$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.1511e+0 2.8316e+0 3.1452e+0 1.1782e+0 Level 2 1.6608e+0 0.9240 3.4495e+0 -0.2848 1.6225e+0 0.9549 6.7356e-1 0.8067 Level 3 9.6287e-1 0.7865 3.7937e+0 -0.1372 8.5141e-1 0.9303 3.5846e-1 0.9100 Level 4 6.7713e-1 0.5079 3.9819e+0 -0.0699 4.5318e-1 0.9098 1.8459e-1 0.9575 Level 5 5.8043e-1 0.2223 4.0839e+0 -0.0365 2.4081e-1 0.9122 9.3619e-2 0.9795
Example 1 - piecewise linear elements (k = 1)
 Mesh $\beta_{0}=3$ $\beta_{0}=4$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.1387e+0 6.5647e-1 3.1321e+0 5.0657e-1 Level 2 1.5825e+0 0.9880 2.0026e-1 1.7129 1.5728e+0 0.9938 1.2993e-1 1.9630 Level 3 7.8972e-1 1.0028 5.5290e-2 1.8568 7.8710e-1 0.9987 3.2384e-2 2.0044 Level 4 3.9408e-1 1.0029 1.4490e-2 1.9320 3.9366e-1 0.9996 8.0207e-3 2.0135 Level 5 1.9690e-1 1.0010 3.7056e-3 1.9673 1.9685e-1 0.9999 1.9895e-3 2.0113
 Mesh $\beta_{0}=3$ $\beta_{0}=4$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.1387e+0 6.5647e-1 3.1321e+0 5.0657e-1 Level 2 1.5825e+0 0.9880 2.0026e-1 1.7129 1.5728e+0 0.9938 1.2993e-1 1.9630 Level 3 7.8972e-1 1.0028 5.5290e-2 1.8568 7.8710e-1 0.9987 3.2384e-2 2.0044 Level 4 3.9408e-1 1.0029 1.4490e-2 1.9320 3.9366e-1 0.9996 8.0207e-3 2.0135 Level 5 1.9690e-1 1.0010 3.7056e-3 1.9673 1.9685e-1 0.9999 1.9895e-3 2.0113
Example 1 - piecewise quadratic elements (k = 2)
 Mesh $\beta_{0}=2$ $\beta_{0}=3$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 4.1581e-1 8.9233e-1 3.7338e-1 3.1242e-1 Level 2 3.8786e-1 0.1004 6.0156e-1 0.5689 1.9600e-1 0.9298 1.0976e-1 1.5091 Level 3 3.1811e-1 0.2860 3.3979e-1 0.8241 7.1110e-2 1.4627 3.1687e-2 1.7924 Level 4 2.2423e-1 0.5045 1.7980e-1 0.9183 1.9708e-2 1.8513 8.4505e-3 1.9068 Level 5 1.3977e-1 0.6819 9.2408e-2 0.9603 4.9833e-3 1.9836 2.1778e-3 1.9561
 Mesh $\beta_{0}=2$ $\beta_{0}=3$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 4.1581e-1 8.9233e-1 3.7338e-1 3.1242e-1 Level 2 3.8786e-1 0.1004 6.0156e-1 0.5689 1.9600e-1 0.9298 1.0976e-1 1.5091 Level 3 3.1811e-1 0.2860 3.3979e-1 0.8241 7.1110e-2 1.4627 3.1687e-2 1.7924 Level 4 2.2423e-1 0.5045 1.7980e-1 0.9183 1.9708e-2 1.8513 8.4505e-3 1.9068 Level 5 1.3977e-1 0.6819 9.2408e-2 0.9603 4.9833e-3 1.9836 2.1778e-3 1.9561
Example 1 - piecewise quadratic elements (k = 2)
 Mesh $\beta_{0}=4$ $\beta_{0}=5$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.2065e-1 1.1501e-1 2.8633e-1 5.0175e-2 Level 2 8.8722e-2 1.8536 2.1377e-2 2.4276 6.8564e-2 2.0622 5.6733e-3 3.1447 Level 3 1.8873e-2 2.2330 3.1356e-3 2.7692 1.6740e-2 2.0342 6.0229e-4 3.2357 Level 4 4.3171e-3 2.1282 4.2411e-4 2.8862 4.1755e-3 2.0033 7.0870e-5 3.0872 Level 5 1.0521e-3 2.0367 5.6265e-5 2.9141 1.0435e-3 2.0005 9.2001e-6 2.9454
 Mesh $\beta_{0}=4$ $\beta_{0}=5$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.2065e-1 1.1501e-1 2.8633e-1 5.0175e-2 Level 2 8.8722e-2 1.8536 2.1377e-2 2.4276 6.8564e-2 2.0622 5.6733e-3 3.1447 Level 3 1.8873e-2 2.2330 3.1356e-3 2.7692 1.6740e-2 2.0342 6.0229e-4 3.2357 Level 4 4.3171e-3 2.1282 4.2411e-4 2.8862 4.1755e-3 2.0033 7.0870e-5 3.0872 Level 5 1.0521e-3 2.0367 5.6265e-5 2.9141 1.0435e-3 2.0005 9.2001e-6 2.9454
Example 2 - piecewise linear elements (k = 1)
 Mesh $\beta_{0}=1$ $\beta_{0}=2$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 4.7884e+1 2.0296e+1 3.8149e+1 5.4663e+0 Level 2 4.0636e+1 0.2367 2.5457e+1 -0.3268 2.4720e+1 0.6259 3.0826e+0 0.8264 Level 3 3.8939e+1 0.0615 2.8694e+1 -0.1726 1.8248e+1 0.4379 1.6804e+0 0.8753 Level 4 3.8678e+1 0.0097 3.0078e+1 -0.0679 1.4554e+1 0.3263 8.7292e-1 0.9448 Level 5 3.8709e+1 -0.0011 3.0800e+1 -0.0342 1.1453e+1 0.3456 4.4531e-1 0.9710
 Mesh $\beta_{0}=1$ $\beta_{0}=2$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 4.7884e+1 2.0296e+1 3.8149e+1 5.4663e+0 Level 2 4.0636e+1 0.2367 2.5457e+1 -0.3268 2.4720e+1 0.6259 3.0826e+0 0.8264 Level 3 3.8939e+1 0.0615 2.8694e+1 -0.1726 1.8248e+1 0.4379 1.6804e+0 0.8753 Level 4 3.8678e+1 0.0097 3.0078e+1 -0.0679 1.4554e+1 0.3263 8.7292e-1 0.9448 Level 5 3.8709e+1 -0.0011 3.0800e+1 -0.0342 1.1453e+1 0.3456 4.4531e-1 0.9710
Example 2 - piecewise linear elements (k = 1)
 Mesh $\beta_{0}=3$ $\beta_{0}=4$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.4092e+1 2.6703e+0 3.2109e+1 2.3053e+0 Level 2 1.8272e+1 0.8997 7.4222e-1 1.8470 1.5700e+1 1.0322 5.9375e-1 1.9570 Level 3 8.9585e+0 1.0283 1.9670e-1 1.9158 7.7869e+0 1.0116 1.4946e-1 1.9900 Level 4 4.1469e+0 1.1112 5.0081e-2 1.9736 3.8903e+0 1.0011 3.7104e-2 2.0101 Level 5 1.9837e+0 1.0638 1.2633e-2 1.9870 1.9450e+0 1.0001 9.2367e-3 2.0061
 Mesh $\beta_{0}=3$ $\beta_{0}=4$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.4092e+1 2.6703e+0 3.2109e+1 2.3053e+0 Level 2 1.8272e+1 0.8997 7.4222e-1 1.8470 1.5700e+1 1.0322 5.9375e-1 1.9570 Level 3 8.9585e+0 1.0283 1.9670e-1 1.9158 7.7869e+0 1.0116 1.4946e-1 1.9900 Level 4 4.1469e+0 1.1112 5.0081e-2 1.9736 3.8903e+0 1.0011 3.7104e-2 2.0101 Level 5 1.9837e+0 1.0638 1.2633e-2 1.9870 1.9450e+0 1.0001 9.2367e-3 2.0061
Example 2 - piecewise quadratic elements (k = 2)
 Mesh $\beta_{0}=2$ $\beta_{0}=3$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 2.1301e+1 4.1219e+0 1.3184e+1 8.4138e-1 Level 2 1.8912e+1 0.1716 2.7544e+0 0.5815 9.4371e+0 0.4823 2.7215e-1 1.6283 Level 3 1.6360e+1 0.2091 1.5988e+0 0.7847 4.4786e+0 1.0752 7.6921e-2 1.8229 Level 4 1.3958e+1 0.2290 8.5329e-1 0.9058 1.4540e+0 1.6230 2.0243e-2 1.9259 Level 5 1.1268e+1 0.3088 4.4049e-1 0.9539 3.9315e-1 1.8868 5.1784e-3 1.9668
 Mesh $\beta_{0}=2$ $\beta_{0}=3$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 2.1301e+1 4.1219e+0 1.3184e+1 8.4138e-1 Level 2 1.8912e+1 0.1716 2.7544e+0 0.5815 9.4371e+0 0.4823 2.7215e-1 1.6283 Level 3 1.6360e+1 0.2091 1.5988e+0 0.7847 4.4786e+0 1.0752 7.6921e-2 1.8229 Level 4 1.3958e+1 0.2290 8.5329e-1 0.9058 1.4540e+0 1.6230 2.0243e-2 1.9259 Level 5 1.1268e+1 0.3088 4.4049e-1 0.9539 3.9315e-1 1.8868 5.1784e-3 1.9668
Example 2 - piecewise quadratic elements (k = 2)
 Mesh $\beta_{0}=4$ $\beta_{0}=5$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 7.7496e+0 1.7252e-1 3.2231e+0 7.4422e-2 Level 2 2.1684e+0 1.8374 2.6866e-2 2.6829 4.0304e-1 2.9994 8.8020e-3 3.0798 Level 3 3.2927e-1 2.7192 3.6813e-3 2.8674 8.2416e-2 2.2899 1.2434e-3 2.8235 Level 4 4.7524e-2 2.7925 5.0460e-4 2.8670 2.4217e-2 1.7669 3.6017e-4 1.7875 Level 5 1.0420e-2 2.1893 1.5692e-4 1.6851 9.0684e-3 1.4171 1.6761e-4 1.1035
 Mesh $\beta_{0}=4$ $\beta_{0}=5$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 7.7496e+0 1.7252e-1 3.2231e+0 7.4422e-2 Level 2 2.1684e+0 1.8374 2.6866e-2 2.6829 4.0304e-1 2.9994 8.8020e-3 3.0798 Level 3 3.2927e-1 2.7192 3.6813e-3 2.8674 8.2416e-2 2.2899 1.2434e-3 2.8235 Level 4 4.7524e-2 2.7925 5.0460e-4 2.8670 2.4217e-2 1.7669 3.6017e-4 1.7875 Level 5 1.0420e-2 2.1893 1.5692e-4 1.6851 9.0684e-3 1.4171 1.6761e-4 1.1035
Example 3 - piecewise linear elements (k = 1)
 Mesh $\beta_{0}=1$ $\beta_{0}=2$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 2.1520e+0 2.9567e+0 2.0507e+0 5.9099e-1 Level 2 1.5717e+0 0.4534 3.2200e+0 -0.1231 1.2779e+0 0.6823 3.4432e-1 0.7794 Level 3 1.2871e+0 0.2882 3.4488e+0 -0.0990 7.5647e-1 0.7564 1.7768e-1 0.9545 Level 4 1.1948e+0 0.1074 3.5461e+0 -0.0401 4.4426e-1 0.7679 8.9088e-2 0.9960 Level 5 1.1645e+0 0.0370 3.5926e+0 -0.0187 2.5595e-1 0.7955 4.4926e-2 0.9876
 Mesh $\beta_{0}=1$ $\beta_{0}=2$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 2.1520e+0 2.9567e+0 2.0507e+0 5.9099e-1 Level 2 1.5717e+0 0.4534 3.2200e+0 -0.1231 1.2779e+0 0.6823 3.4432e-1 0.7794 Level 3 1.2871e+0 0.2882 3.4488e+0 -0.0990 7.5647e-1 0.7564 1.7768e-1 0.9545 Level 4 1.1948e+0 0.1074 3.5461e+0 -0.0401 4.4426e-1 0.7679 8.9088e-2 0.9960 Level 5 1.1645e+0 0.0370 3.5926e+0 -0.0187 2.5595e-1 0.7955 4.4926e-2 0.9876
Example 3 - piecewise linear elements (k = 1)
 Mesh $\beta_{0}=3$ $\beta_{0}=4$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 1.9345e+0 1.9955e-1 1.8922e+0 1.4389e-1 Level 2 1.0109e+0 0.9363 6.4142e-2 1.6374 9.8873e-1 0.9364 4.0633e-2 1.8242 Level 3 4.9094e-1 1.0420 1.6236e-2 1.9821 4.8750e-1 1.0202 9.6646e-3 2.0719 Level 4 2.4285e-1 1.0155 4.0744e-3 1.9945 2.4244e-1 1.0078 2.3708e-3 2.0273 Level 5 1.2177e-1 0.9959 1.0372e-3 1.9738 1.2172e-1 0.9940 5.9750e-4 1.9883
 Mesh $\beta_{0}=3$ $\beta_{0}=4$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 1.9345e+0 1.9955e-1 1.8922e+0 1.4389e-1 Level 2 1.0109e+0 0.9363 6.4142e-2 1.6374 9.8873e-1 0.9364 4.0633e-2 1.8242 Level 3 4.9094e-1 1.0420 1.6236e-2 1.9821 4.8750e-1 1.0202 9.6646e-3 2.0719 Level 4 2.4285e-1 1.0155 4.0744e-3 1.9945 2.4244e-1 1.0078 2.3708e-3 2.0273 Level 5 1.2177e-1 0.9959 1.0372e-3 1.9738 1.2172e-1 0.9940 5.9750e-4 1.9883
Example 3 - piecewise quadratic elements (k = 2)
 Mesh $\beta_{0}=4$ $\beta_{0}=5$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 2.0263e-1 2.1517e-2 1.2596e-1 7.5252e-3 Level 2 4.6043e-2 2.1377 4.0745e-3 2.4007 3.1587e-2 1.9955 1.0519e-3 2.8387 Level 3 8.9009e-3 2.3709 6.1530e-4 2.7272 8.0377e-3 1.9744 2.0217e-4 2.3793 Level 4 2.2145e-3 2.0069 1.1113e-4 2.4690 2.1738e-3 1.8865 5.8895e-5 1.7793 Level 5 6.0543e-4 1.8709 2.6078e-5 2.0913 6.0306e-4 1.8498 1.9505e-5 1.5943
 Mesh $\beta_{0}=4$ $\beta_{0}=5$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 2.0263e-1 2.1517e-2 1.2596e-1 7.5252e-3 Level 2 4.6043e-2 2.1377 4.0745e-3 2.4007 3.1587e-2 1.9955 1.0519e-3 2.8387 Level 3 8.9009e-3 2.3709 6.1530e-4 2.7272 8.0377e-3 1.9744 2.0217e-4 2.3793 Level 4 2.2145e-3 2.0069 1.1113e-4 2.4690 2.1738e-3 1.8865 5.8895e-5 1.7793 Level 5 6.0543e-4 1.8709 2.6078e-5 2.0913 6.0306e-4 1.8498 1.9505e-5 1.5943
Example 3 - piecewise linear elements (k = 1), $\beta_0 = 3$
 Mesh $A_1=1, A_2=10$ $A_1=1, A_2=1000$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.4237e-1 2.1877e-2 1.9445e+1 2.0541e+0 Level 2 1.5292e-1 1.1628 5.8414e-3 1.9050 8.7537e+0 1.1514 5.3965e-1 1.9284 Level 3 7.2010e-2 1.0865 1.5141e-3 1.9479 4.4078e+0 0.9898 1.4652e-1 1.8809 Level 4 3.5078e-2 1.0376 3.8096e-4 1.9907 1.6765e+0 1.3946 3.7435e-2 1.9686 Level 5 1.7351e-2 1.0155 9.6691e-5 1.9782 5.0627e-1 1.7275 9.6667e-3 1.9533
 Mesh $A_1=1, A_2=10$ $A_1=1, A_2=1000$ $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order $||| e_{h}|||$ Order $\Vert e_{0}\Vert$ Order Level 1 3.4237e-1 2.1877e-2 1.9445e+1 2.0541e+0 Level 2 1.5292e-1 1.1628 5.8414e-3 1.9050 8.7537e+0 1.1514 5.3965e-1 1.9284 Level 3 7.2010e-2 1.0865 1.5141e-3 1.9479 4.4078e+0 0.9898 1.4652e-1 1.8809 Level 4 3.5078e-2 1.0376 3.8096e-4 1.9907 1.6765e+0 1.3946 3.7435e-2 1.9686 Level 5 1.7351e-2 1.0155 9.6691e-5 1.9782 5.0627e-1 1.7275 9.6667e-3 1.9533
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