# American Institute of Mathematical Sciences

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  May 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
##### References:
 [1] 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. [2] 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. [3] 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. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] 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. [23] J. Wang and X. Ye, The basics of weak Galerkin finite element methods, preprint, arXiv: 1901.10035. [24] 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.

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##### References:
 [1] 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. [2] 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. [3] 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. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] 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. [23] J. Wang and X. Ye, The basics of weak Galerkin finite element methods, preprint, arXiv: 1901.10035. [24] 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.
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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