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Volume 23, Issue 1
Error Estimates for the Iterative Discontinuous Galerkin Method to the Nonlinear Poisson-Boltzmann Equation

Peimeng Yin, Yunqing Huang & Hailiang Liu

DOI: 10.4208/cicp.OA-2016-0226

Commun. Comput. Phys., 23 (2018), pp. 168-197.

Published online: 2018-01

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  • Abstract

This paper is devoted to the error estimate for the iterative discontinuous Galerkin (IDG) method introduced in [P. Yin, Y. Huang and H. Liu. Commun. Comput. Phys. 16: 491–515, 2014] to the nonlinear Poisson-Boltzmann equation. The total error includes both the iteration error and the discretization error of the direct DG method to linear elliptic equations. For the DDG method, the energy error is obtained by a constructive approach through an explicit global projection satisfying interface conditions dictated by the choice of numerical fluxes. The $L^2$ error of order O(hm+1) for polynomials of degree m is further recovered. The bounding constant is also shown to be independent of the iteration times. Numerical tests are given to validate the established convergence theory.

  • Keywords

Poisson-Boltzmann equation, DG methods, global projection, energy error estimates, $L^2$ error estimates.

  • AMS Subject Headings

65D15, 65N30, 35J05, 35J25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-23-168, author = {Peimeng Yin, Yunqing Huang and Hailiang Liu}, title = {Error Estimates for the Iterative Discontinuous Galerkin Method to the Nonlinear Poisson-Boltzmann Equation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {23}, number = {1}, pages = {168--197}, abstract = {

This paper is devoted to the error estimate for the iterative discontinuous Galerkin (IDG) method introduced in [P. Yin, Y. Huang and H. Liu. Commun. Comput. Phys. 16: 491–515, 2014] to the nonlinear Poisson-Boltzmann equation. The total error includes both the iteration error and the discretization error of the direct DG method to linear elliptic equations. For the DDG method, the energy error is obtained by a constructive approach through an explicit global projection satisfying interface conditions dictated by the choice of numerical fluxes. The $L^2$ error of order O(hm+1) for polynomials of degree m is further recovered. The bounding constant is also shown to be independent of the iteration times. Numerical tests are given to validate the established convergence theory.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0226}, url = {http://global-sci.org/intro/article_detail/cicp/10524.html} }
TY - JOUR T1 - Error Estimates for the Iterative Discontinuous Galerkin Method to the Nonlinear Poisson-Boltzmann Equation AU - Peimeng Yin, Yunqing Huang & Hailiang Liu JO - Communications in Computational Physics VL - 1 SP - 168 EP - 197 PY - 2018 DA - 2018/01 SN - 23 DO - http://doi.org/10.4208/cicp.OA-2016-0226 UR - https://global-sci.org/intro/article_detail/cicp/10524.html KW - Poisson-Boltzmann equation, DG methods, global projection, energy error estimates, $L^2$ error estimates. AB -

This paper is devoted to the error estimate for the iterative discontinuous Galerkin (IDG) method introduced in [P. Yin, Y. Huang and H. Liu. Commun. Comput. Phys. 16: 491–515, 2014] to the nonlinear Poisson-Boltzmann equation. The total error includes both the iteration error and the discretization error of the direct DG method to linear elliptic equations. For the DDG method, the energy error is obtained by a constructive approach through an explicit global projection satisfying interface conditions dictated by the choice of numerical fluxes. The $L^2$ error of order O(hm+1) for polynomials of degree m is further recovered. The bounding constant is also shown to be independent of the iteration times. Numerical tests are given to validate the established convergence theory.

Peimeng Yin, Yunqing Huang and Hailiang Liu. (2018). Error Estimates for the Iterative Discontinuous Galerkin Method to the Nonlinear Poisson-Boltzmann Equation. Communications in Computational Physics. 23 (1). 168-197. doi:10.4208/cicp.OA-2016-0226
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