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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 |
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.
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. |
show all references
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. |







Mesh | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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 | ||||||||
Order | Order | Order | 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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