
-
Previous Article
Analysis of strategic customer behavior in fuzzy queueing systems
- JIMO Home
- This Issue
-
Next Article
Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax
A simple and efficient technique to accelerate the computation of a nonlocal dielectric model for electrostatics of biomolecule
1. | School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114 China |
2. | School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083 China |
The nonlocal modified Poisson-Boltzmann equation (NMPBE) is one important variant of a commonly-used dielectric continuum model, Poisson-Boltzmann equation (PBE). In this paper, we use a nonlinear block relaxation method to develop a new nonlinear solver for the nonlinear equation of $\Phi $ and thus a new NMPBE solver, which is then programmed as a software package in $\texttt{C}\backslash\texttt{C++}$, $\texttt{Fortran}$ and $\texttt{Python}$ for computing the electrostatics of a protein in a symmetric 1:1 ionic solvent. Numerical tests validate the new package and show that the new solver can improve the efficiency by at least $ 40\%$ than the finite element NMPBE solver without compromising solution accuracy.
References:
[1] |
M. V. Basilevsky and D. F. Parsons,
An advanced continuum medium model for treating solvation effects: Nonlocal electrostatics with a cavity, The Journal of Chemical Physics, 105 (1996), 3734-3746.
doi: 10.1063/1.472193. |
[2] |
I. Borukhov, D. Andelman and H. Orland, Steric effects in electrolytes: A modified Poisson-Boltzmann equation, Physical Review Letters, 79 (1997), 435.
doi: 10.1103/PhysRevLett.79.435. |
[3] |
V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag and S. Doniach,
Evaluation of ion binding to DNA duplexes using a size-modified Poisson-Boltzmann theory, Biophysical Journal, 93 (2007), 3202-3209.
doi: 10.1529/biophysj.106.099168. |
[4] |
W. Deng, J. Xu and S. Zhao,
On developing stable finite element methods for pseudo-time simulation of biomolecular electrostatics, Journal of Computational and Applied Mathematics, 330 (2018), 456-474.
doi: 10.1016/j.cam.2017.09.004. |
[5] |
R. Dogonadze and A. Kornyshev,
Polar solvent structure in the theory of ionic solvation, Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics, 70 (1974), 1121-1132.
doi: 10.1039/f29747001121. |
[6] |
A. Hildebrandt, R. Blossey, S. Rjasanow, O. Kohlbacher and H.-P. Lenhof, Novel formulation of nonlocal electrostatics, Physical Review Letters, 93 (2004), 108104. Google Scholar |
[7] |
B. Honig and A. Nicholls,
Classical electrostatics in biology and chemistry, Science, 268 (1995), 1144-1149.
doi: 10.1126/science.7761829. |
[8] |
J. Hu, S. Zhao and W. Geng, Accurate pKa computation using matched interface and boundary (MIB) method based Poisson-Boltzmann solver, Communication in Computational Physics, 23 (2018), 520-539. Google Scholar |
[9] |
A. Kornyshev, A. Rubinshtein and M. Vorotyntsev, Model nonlocal electrostatics. Ⅰ, Journal of Physics C: Solid State Physics, 11 (1978), 3307.
doi: 10.1088/0022-3719/11/15/029. |
[10] |
A. A. Kornyshev and G. Sutmann, Nonlocal dielectric saturation in liquid water, Physical Review Letters, 79 (1997), 3435.
doi: 10.1103/PhysRevLett.79.3435. |
[11] |
J. Li and D. Xie,
A new linear Poisson-Boltzmann equation and finite element solver by solution decomposition approach, Communications in Mathematical Sciences, 13 (2015), 315-325.
doi: 10.4310/CMS.2015.v13.n2.a2. |
[12] |
J. Li and D. Xie,
An effective minimization protocol for solving a size-modified Poisson-Boltzmann equation for biomolecule in ionic solvent, International Journal of Numerical Analysis and Modeling, 12 (2015), 286-301.
|
[13] |
J. Li, J. Ying and D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Analysis: Real World Applications, submitted. Google Scholar |
[14] |
J. L. Liu,
Numerical methods for the Poisson-Fermi equation in electrolytes, Journal of Computational Physics, 247 (2013), 88-99.
doi: 10.1016/j.jcp.2013.03.058. |
[15] |
A. Logg and G. N. Wells, DOLFIN: Automated finite element computing, ACM Transactions on Mathematical Software (TOMS), 37 (2010), Art. 20, 28 pp.
doi: 10.1145/1731022.1731030. |
[16] |
L. R. Scott and D. Xie, Analysis of a Nonlocal Poisson-Boltzmann Equation, Technical report, Research Report UC/CS TR-2016-1, Dept. Comp. Sci., Univ. Chicago, 2016. Google Scholar |
[17] |
M. Vorotyntsev, Model nonlocal electrostatics. Ⅱ. spherical interface, Journal of Physics C: Solid State Physics, 11 (1978), 3323.
doi: 10.1088/0022-3719/11/15/030. |
[18] |
S. Weggler, V. Rutka and A. Hildebrandt,
A new numerical method for nonlocal electrostatics in biomolecular simulations, Journal of Computational Physics, 229 (2010), 4059-4074.
doi: 10.1016/j.jcp.2010.01.040. |
[19] |
D. Xie and Y. Jiang,
A nonlocal modified Poisson-Boltzmann equation and finite element solver for computing electrostatics of biomolecules, Journal of Computational Physics, 322 (2016), 1-20.
doi: 10.1016/j.jcp.2016.06.028. |
[20] |
D. Xie, Y. Jiang and L. R. Scott,
Efficient algorithms for a nonlocal dielectric model for protein in ionic solvent, SIAM Journal on Scientific Computing, 35 (2013), B1267-B1284.
doi: 10.1137/120899078. |
[21] |
D. Xie and J. Li,
A new analysis of electrostatic free energy minimization and Poisson-Boltzmann equation for protein in ionic solvent, Nonlinear Analysis: Real World Applications, 21 (2015), 185-196.
doi: 10.1016/j.nonrwa.2014.07.008. |
[22] |
D. Xie, H. W. Volkmer and J. Ying, Analytical solutions of nonlocal poisson dielectric models with multiple point charges inside a dielectric sphere, Physical Review E, 93 (2016), 043304.
doi: 10.1103/PhysRevE.93.043304. |
[23] |
D. Xie and J. Ying,
A new box iterative method for a class of nonlinear interface problems with application in solving Poisson-Boltzmann equation, Journal of Computational and Applied Mathematics, 307 (2016), 319-334.
doi: 10.1016/j.cam.2016.01.005. |
[24] |
J. Ying and D. Xie,
A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule, Journal of Computational Physics, 298 (2015), 636-651.
doi: 10.1016/j.jcp.2015.06.016. |
[25] |
J. Ying and D. Xie,
A hybrid solver of size modified Poisson-Boltzmann equation by domain decomposition, finite element, and finite difference, Applied Mathematical Modelling, 58 (2018), 166-180.
doi: 10.1016/j.apm.2017.09.026. |
show all references
References:
[1] |
M. V. Basilevsky and D. F. Parsons,
An advanced continuum medium model for treating solvation effects: Nonlocal electrostatics with a cavity, The Journal of Chemical Physics, 105 (1996), 3734-3746.
doi: 10.1063/1.472193. |
[2] |
I. Borukhov, D. Andelman and H. Orland, Steric effects in electrolytes: A modified Poisson-Boltzmann equation, Physical Review Letters, 79 (1997), 435.
doi: 10.1103/PhysRevLett.79.435. |
[3] |
V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag and S. Doniach,
Evaluation of ion binding to DNA duplexes using a size-modified Poisson-Boltzmann theory, Biophysical Journal, 93 (2007), 3202-3209.
doi: 10.1529/biophysj.106.099168. |
[4] |
W. Deng, J. Xu and S. Zhao,
On developing stable finite element methods for pseudo-time simulation of biomolecular electrostatics, Journal of Computational and Applied Mathematics, 330 (2018), 456-474.
doi: 10.1016/j.cam.2017.09.004. |
[5] |
R. Dogonadze and A. Kornyshev,
Polar solvent structure in the theory of ionic solvation, Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics, 70 (1974), 1121-1132.
doi: 10.1039/f29747001121. |
[6] |
A. Hildebrandt, R. Blossey, S. Rjasanow, O. Kohlbacher and H.-P. Lenhof, Novel formulation of nonlocal electrostatics, Physical Review Letters, 93 (2004), 108104. Google Scholar |
[7] |
B. Honig and A. Nicholls,
Classical electrostatics in biology and chemistry, Science, 268 (1995), 1144-1149.
doi: 10.1126/science.7761829. |
[8] |
J. Hu, S. Zhao and W. Geng, Accurate pKa computation using matched interface and boundary (MIB) method based Poisson-Boltzmann solver, Communication in Computational Physics, 23 (2018), 520-539. Google Scholar |
[9] |
A. Kornyshev, A. Rubinshtein and M. Vorotyntsev, Model nonlocal electrostatics. Ⅰ, Journal of Physics C: Solid State Physics, 11 (1978), 3307.
doi: 10.1088/0022-3719/11/15/029. |
[10] |
A. A. Kornyshev and G. Sutmann, Nonlocal dielectric saturation in liquid water, Physical Review Letters, 79 (1997), 3435.
doi: 10.1103/PhysRevLett.79.3435. |
[11] |
J. Li and D. Xie,
A new linear Poisson-Boltzmann equation and finite element solver by solution decomposition approach, Communications in Mathematical Sciences, 13 (2015), 315-325.
doi: 10.4310/CMS.2015.v13.n2.a2. |
[12] |
J. Li and D. Xie,
An effective minimization protocol for solving a size-modified Poisson-Boltzmann equation for biomolecule in ionic solvent, International Journal of Numerical Analysis and Modeling, 12 (2015), 286-301.
|
[13] |
J. Li, J. Ying and D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Analysis: Real World Applications, submitted. Google Scholar |
[14] |
J. L. Liu,
Numerical methods for the Poisson-Fermi equation in electrolytes, Journal of Computational Physics, 247 (2013), 88-99.
doi: 10.1016/j.jcp.2013.03.058. |
[15] |
A. Logg and G. N. Wells, DOLFIN: Automated finite element computing, ACM Transactions on Mathematical Software (TOMS), 37 (2010), Art. 20, 28 pp.
doi: 10.1145/1731022.1731030. |
[16] |
L. R. Scott and D. Xie, Analysis of a Nonlocal Poisson-Boltzmann Equation, Technical report, Research Report UC/CS TR-2016-1, Dept. Comp. Sci., Univ. Chicago, 2016. Google Scholar |
[17] |
M. Vorotyntsev, Model nonlocal electrostatics. Ⅱ. spherical interface, Journal of Physics C: Solid State Physics, 11 (1978), 3323.
doi: 10.1088/0022-3719/11/15/030. |
[18] |
S. Weggler, V. Rutka and A. Hildebrandt,
A new numerical method for nonlocal electrostatics in biomolecular simulations, Journal of Computational Physics, 229 (2010), 4059-4074.
doi: 10.1016/j.jcp.2010.01.040. |
[19] |
D. Xie and Y. Jiang,
A nonlocal modified Poisson-Boltzmann equation and finite element solver for computing electrostatics of biomolecules, Journal of Computational Physics, 322 (2016), 1-20.
doi: 10.1016/j.jcp.2016.06.028. |
[20] |
D. Xie, Y. Jiang and L. R. Scott,
Efficient algorithms for a nonlocal dielectric model for protein in ionic solvent, SIAM Journal on Scientific Computing, 35 (2013), B1267-B1284.
doi: 10.1137/120899078. |
[21] |
D. Xie and J. Li,
A new analysis of electrostatic free energy minimization and Poisson-Boltzmann equation for protein in ionic solvent, Nonlinear Analysis: Real World Applications, 21 (2015), 185-196.
doi: 10.1016/j.nonrwa.2014.07.008. |
[22] |
D. Xie, H. W. Volkmer and J. Ying, Analytical solutions of nonlocal poisson dielectric models with multiple point charges inside a dielectric sphere, Physical Review E, 93 (2016), 043304.
doi: 10.1103/PhysRevE.93.043304. |
[23] |
D. Xie and J. Ying,
A new box iterative method for a class of nonlinear interface problems with application in solving Poisson-Boltzmann equation, Journal of Computational and Applied Mathematics, 307 (2016), 319-334.
doi: 10.1016/j.cam.2016.01.005. |
[24] |
J. Ying and D. Xie,
A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule, Journal of Computational Physics, 298 (2015), 636-651.
doi: 10.1016/j.jcp.2015.06.016. |
[25] |
J. Ying and D. Xie,
A hybrid solver of size modified Poisson-Boltzmann equation by domain decomposition, finite element, and finite difference, Applied Mathematical Modelling, 58 (2018), 166-180.
doi: 10.1016/j.apm.2017.09.026. |

Index | PDB ID | Index | PDB ID | ||
1 | 2LZX | 488 | 7 | 1A63 | 2065 |
2 | 1AJJ | 513 | 8 | 1CID | 2783 |
3 | 1FXD | 811 | 9 | 1A7M | 2803 |
4 | 1HPT | 852 | 10 | 2AQ5 | 6024 |
5 | 4PTI | 892 | 11 | 1F6W | 8243 |
6 | 1SVR | 1433 | 12 | 1C4K | 11439 |
Index | PDB ID | Index | PDB ID | ||
1 | 2LZX | 488 | 7 | 1A63 | 2065 |
2 | 1AJJ | 513 | 8 | 1CID | 2783 |
3 | 1FXD | 811 | 9 | 1A7M | 2803 |
4 | 1HPT | 852 | 10 | 2AQ5 | 6024 |
5 | 4PTI | 892 | 11 | 1F6W | 8243 |
6 | 1SVR | 1433 | 12 | 1C4K | 11439 |
PDB ID | Number of Mesh Nodes | Iter. Number | Find | Total Time | Relative error | Residual norm | ||
New | FE | New | FE | |||||
2LZX | 26349 | 11 | 15.23 | 27.79 | 27.61 | 40.17 | ||
1AJJ | 31910 | 11 | 26.60 | 48.25 | 45.29 | 66.94 | ||
1FXD | 34469 | 12 | 23.19 | 42.48 | 49.74 | 69.03 | ||
1HPT | 48229 | 10 | 32.33 | 58.78 | 58.08 | 84.53 | ||
4PTI | 39468 | 10 | 25.85 | 46.52 | 47.14 | 67.81 | ||
1SVR | 61074 | 11 | 55.17 | 90.59 | 96.88 | 132.30 | ||
1A63 | 22054 | 11 | 13.52 | 27.09 | 27.19 | 40.76 | ||
1CID | 19872 | 10 | 11.07 | 21.68 | 23.10 | 33.71 | ||
1A7M | 20883 | 10 | 11.63 | 22.42 | 24.53 | 35.33 | ||
2AQ5 | 38151 | 11 | 29.58 | 53.69 | 69.88 | 93.99 | ||
1F6W | 49006 | 11 | 46.77 | 86.41 | 94.47 | 134.11 | ||
1C4K | 72046 | 11 | 70.04 | 118.93 | 172.47 | 221.36 |
PDB ID | Number of Mesh Nodes | Iter. Number | Find | Total Time | Relative error | Residual norm | ||
New | FE | New | FE | |||||
2LZX | 26349 | 11 | 15.23 | 27.79 | 27.61 | 40.17 | ||
1AJJ | 31910 | 11 | 26.60 | 48.25 | 45.29 | 66.94 | ||
1FXD | 34469 | 12 | 23.19 | 42.48 | 49.74 | 69.03 | ||
1HPT | 48229 | 10 | 32.33 | 58.78 | 58.08 | 84.53 | ||
4PTI | 39468 | 10 | 25.85 | 46.52 | 47.14 | 67.81 | ||
1SVR | 61074 | 11 | 55.17 | 90.59 | 96.88 | 132.30 | ||
1A63 | 22054 | 11 | 13.52 | 27.09 | 27.19 | 40.76 | ||
1CID | 19872 | 10 | 11.07 | 21.68 | 23.10 | 33.71 | ||
1A7M | 20883 | 10 | 11.63 | 22.42 | 24.53 | 35.33 | ||
2AQ5 | 38151 | 11 | 29.58 | 53.69 | 69.88 | 93.99 | ||
1F6W | 49006 | 11 | 46.77 | 86.41 | 94.47 | 134.11 | ||
1C4K | 72046 | 11 | 70.04 | 118.93 | 172.47 | 221.36 |
PDB ID | Number of Mesh Nodes | Iter. Number | Find | Total Time | Relative error | Residual norm | ||
New | FE | New | FE | |||||
2LZX | 535400 | 10 | 167.0 | 291.2 | 311.2 | 435.3 | ||
1AJJ | 538321 | 10 | 223.9 | 437.3 | 387.4 | 600.8 | ||
1FXD | 540849 | 11 | 201.5 | 346.2 | 363.1 | 507.8 | ||
1HPT | 543220 | 9 | 186.8 | 399.6 | 332.3 | 545.0 | ||
4PTI | 541329 | 9 | 173.4 | 328.7 | 319.5 | 474.7 | ||
1SVR | 550170 | 10 | 229.7 | 411.0 | 415.3 | 596.7 | ||
1A63 | 558010 | 11 | 253.3 | 442.8 | 573.1 | 762.7 | ||
1CID | 558374 | 10 | 203.0 | 389.0 | 409.4 | 595.4 | ||
1A7M | 563919 | 11 | 242.9 | 471.7 | 442.6 | 671.4 | ||
2AQ5 | 577821 | 10 | 296.7 | 566.8 | 637.7 | 907.8 | ||
1F6W | 574686 | 11 | 332.1 | 597.3 | 707.8 | 973.0 | ||
1C4K | 573111 | 14 | 396.6 | 698.3 | 940.8 | 1242.5 |
PDB ID | Number of Mesh Nodes | Iter. Number | Find | Total Time | Relative error | Residual norm | ||
New | FE | New | FE | |||||
2LZX | 535400 | 10 | 167.0 | 291.2 | 311.2 | 435.3 | ||
1AJJ | 538321 | 10 | 223.9 | 437.3 | 387.4 | 600.8 | ||
1FXD | 540849 | 11 | 201.5 | 346.2 | 363.1 | 507.8 | ||
1HPT | 543220 | 9 | 186.8 | 399.6 | 332.3 | 545.0 | ||
4PTI | 541329 | 9 | 173.4 | 328.7 | 319.5 | 474.7 | ||
1SVR | 550170 | 10 | 229.7 | 411.0 | 415.3 | 596.7 | ||
1A63 | 558010 | 11 | 253.3 | 442.8 | 573.1 | 762.7 | ||
1CID | 558374 | 10 | 203.0 | 389.0 | 409.4 | 595.4 | ||
1A7M | 563919 | 11 | 242.9 | 471.7 | 442.6 | 671.4 | ||
2AQ5 | 577821 | 10 | 296.7 | 566.8 | 637.7 | 907.8 | ||
1F6W | 574686 | 11 | 332.1 | 597.3 | 707.8 | 973.0 | ||
1C4K | 573111 | 14 | 396.6 | 698.3 | 940.8 | 1242.5 |
[1] |
Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024 |
[2] |
Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154 |
[3] |
Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109 |
[4] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[5] |
Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems & Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947 |
[6] |
Ruijun Zhao, Yong-Tao Zhang, Shanqin Chen. Krylov implicit integration factor WENO method for SIR model with directed diffusion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4983-5001. doi: 10.3934/dcdsb.2019041 |
[7] |
Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 |
[8] |
Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339 |
[9] |
Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 |
[10] |
Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637 |
[11] |
Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153 |
[12] |
Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665 |
[13] |
Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387 |
[14] |
So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 |
[15] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
[16] |
Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052 |
[17] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[18] |
Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051 |
[19] |
Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297 |
[20] |
Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]