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## Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear Poisson-Boltzmann Equation (1107)

Citations: | 2 - 1 self |

### Citations

204 |
A convergent adaptive algorithm for Poisson’s equation,
- Dorfler
- 1996
(Show Context)
Citation Context ... be exact, namely uk is the exact solution to the nonlinear equation (2); the ESTIMATE routine computes the elementwise residual indicator η(uk,τ); the MARK routine uses standard Dörfler marking (cf. =-=[6]-=-) where Mk ⊂ Tk is chosen so that4 Michael Holst, Ryan Szypowski, and Yunrong Zhu η(uk,Mk) ≥ θη(uk,Tk) for some parameter θ ∈ (0,1]; finally, the routine REFINE subdivide the marked elements and poss... |

89 | Optimality of a standard adaptive finite element method,
- Stevenson
- 2007
(Show Context)
Citation Context ...therein for linear PDE case, and [8] for nonlinear PDE case). To the best of the authors knowledge, there are only a couple of convergence results of AFEM for symmetric linear elliptic equations (cf. =-=[10, 1]-=-) which take the numerical error into account. To distinct with the exact solver case, we use ûk and Tk ˆ to denote the numerical approximation to (2) and the triangulation obtained from the adaptive ... |

65 |
Two-grid discretization techniques for linear and nonlinear PDEs
- Xu
- 1996
(Show Context)
Citation Context ... For a proof of the inequality (10), see for example [7]. To achieve the optimal computational complexity, we should avoid solving the nonlinear system (2) as much as we could. The two-grid algorithm =-=[11]-=- shows that a nonlinear solver on a coarse grid combined with a Newton update on the fine grid still yield quasi-optimal approximation. Motivated by this idea, we propose the following AFEM algorithm ... |

49 |
Theory of adaptive finite element methods: An introduction, Multiscale, Nonlinear and Adaptive Approximation
- Nochetto, Siebert, et al.
- 2009
(Show Context)
Citation Context ...n way such that the new triangulation preserves shape-regularity and conformity. During last decade, a lot of theoretical work has been done to show the convergence of the AFEM with exact solver (see =-=[9]-=- and the references cited therein for linear PDE case, and [8] for nonlinear PDE case). To the best of the authors knowledge, there are only a couple of convergence results of AFEM for symmetric linea... |

31 | The finite element approximation of the nonlinear poissonboltzmann equation
- Chen, Holst, et al.
- 2007
(Show Context)
Citation Context ...stant ε|Ωm = εm and ε|Ωs = εs. The modified Debye-Hückel parameter κ2 is also piecewise constant with κ2 (x)|Ωm = 0 and κ2 (x)|Ωs > 0. The equation (1) arises from several regularization schemes (cf. =-=[4, 5]-=-) of the nonlinear Poisson-Boltzmann equation: −∇ · (ε∇u) + κ 2 sinhu = N ∑ i=1 ziδ(xi), where the right hand side represents N fixed points with charges zi at positions xi, and δ is the Dirac delta d... |

14 | Practical adaptive finite element modeling techniques for the pbe
- Bond, Holst, et al.
- 2007
(Show Context)
Citation Context ...(1) and the discrete solution uh to (2), which play a key role in the finite elementInexact AFEM for Nonlinear PBE 3 error analysis of (2) and adaptive algorithms. For interested reader, we refer to =-=[4, 7]-=- for details. Theorem 1. There exist u+,u− ∈ L ∞ (Ω) such that the solution u of (1) satisfies the following a priori L ∞ bounds: Moreover, if the triangulation Th satisfies that u− ≤ u ≤ u+, a.e. in ... |

11 | Accurate evaluation of electrostatics for macromolecules in solution
- Chern, Liu, et al.
(Show Context)
Citation Context ...stant ε|Ωm = εm and ε|Ωs = εs. The modified Debye-Hückel parameter κ2 is also piecewise constant with κ2 (x)|Ωm = 0 and κ2 (x)|Ωs > 0. The equation (1) arises from several regularization schemes (cf. =-=[4, 5]-=-) of the nonlinear Poisson-Boltzmann equation: −∇ · (ε∇u) + κ 2 sinhu = N ∑ i=1 ziδ(xi), where the right hand side represents N fixed points with charges zi at positions xi, and δ is the Dirac delta d... |

6 | Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids, ArXiv e-prints
- Chen, Holst, et al.
- 2010
(Show Context)
Citation Context ... (b ′ (ûk)(ûk+1 − ûk),φ) = 0 (11) for every φ ∈ V ( ˆ Tk+1). We remark that since (11) is only a linear problem, we could use the local multilevel method to solve it in (near) optimal complexity (cf. =-=[3]-=-). Therefore, the overall computational complexity of the Algorithm 1 is nearly optimal. We should point out that it is not obvioius how to enforce the required approximation property (8) that ûk must... |

6 | Local convergence of adaptive methods for nonlinear partial differential equations. Submitted for publication. Available as arXiv:1001.1382 [math.NA
- Holst, Tsogtgerel, et al.
(Show Context)
Citation Context ...ty and conformity. During last decade, a lot of theoretical work has been done to show the convergence of the AFEM with exact solver (see [9] and the references cited therein for linear PDE case, and =-=[8]-=- for nonlinear PDE case). To the best of the authors knowledge, there are only a couple of convergence results of AFEM for symmetric linear elliptic equations (cf. [10, 1]) which take the numerical er... |

4 |
Convergence of AFEM for semilinear problems with inexact solvers
- Bank, Holst, et al.
- 2011
(Show Context)
Citation Context ...est level, and the remaining levels involve only single Newton updates to the previous approximate solution. We summarize a recently developed AFEM convergence theory for inexact solvers appearing in =-=[2]-=-, and present a sequence of numerical experiments that give evidence that the theory does in fact predict the contraction properties of AFEM with inexact solvers. The various routines used are all des... |

1 |
Convergence of inexact adaptive finite element solvers for elliptic problems
- Arioli, Georgoulis, et al.
- 2009
(Show Context)
Citation Context ...therein for linear PDE case, and [8] for nonlinear PDE case). To the best of the authors knowledge, there are only a couple of convergence results of AFEM for symmetric linear elliptic equations (cf. =-=[10, 1]-=-) which take the numerical error into account. To distinct with the exact solver case, we use ûk and Tk ˆ to denote the numerical approximation to (2) and the triangulation obtained from the adaptive ... |