# zbMATH — the first resource for mathematics

Approximate solutions to Poisson-Boltzmann systems with Sobolev gradients. (English) Zbl 1220.65166
Summary: A weighted Sobolev gradient approach [W. T. Mahavier, Nonlinear World 4, No. 4, 435–455 (1997; Zbl 0908.65060)] is presented to a nonlinear PBE [M. J. Holst, The Poisson-Boltzmann equation: analysis and multilevel numerical solution, Ph.D. thesis, Univ. of Illinois, USA (1994)] with discontinuous coefficient functions. A comparison is given between the weighted and unweighted Sobolev gradient in the finite element setting in two and three dimensions. Behavior of the various Sobolev gradients is discussed for large jump size in the coefficient. A comparison with Newton’s method is given where the failure of Newton’s method is demonstrated for a test problem.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
  Mahavier, W.T., A numerical method utilizing weighted Sobolev descent to solve singular differential equations, Nonlinear world, 4, 435-455, (1997) · Zbl 0908.65060  M.J. Holst, The Poisson-Boltzmann Equation: analysis and multilevel numerical solution. Ph.D. Thesis, Univ. of Illinois, USA, 1994.  Karatson, J.; Farago, I., Preconditioning operators and Sobolev gradients for nonlinear elliptic problems, Comput. math. appl., 50, 1077-1092, (2005) · Zbl 1118.65122  Neuberger, J.W., Sobolev gradient and differential equations, Springer lecture notes in mathematics # 1670, (2010), Springer-Verlag New York  Richardson, W.B., Sobolev preconditioning for the poisson – boltzmann equation, Comput. methods appl. mech. eng., 181, 425-436, (2000) · Zbl 0960.82035  Renka, R.J., Constructing fair curves and surfaces with a Sobolev gradient method, Cagd, 21, 137-149, (2004) · Zbl 1069.65565  Knowles, I., Variational methods for ill-posed problems, Contemp. math., 357, 187-199, (2004) · Zbl 1064.65132  Raza, N.; Sial, S.; Siddiqi, S.S., Sobolev gradient approach for time the evolution related to the energy minimization of Ginzburg-Landau functional, J. comput. phys., 228, 2566-2571, (2009) · Zbl 1166.65348  Sial, S.; Neuberger, J.W.; Lookman, T.; Saxena, A., Energy minimization using phase Sobolev gradient: application to phase separation and ordering, J. comput. phys., 189, 88-97, (2003) · Zbl 1097.49002  Neuberger, J.W.; Renka, R.J., Computational simulation of vortex phenomena in superconductors, Elect. J. differ. equat. conf., 10, 245-250, (2003) · Zbl 1109.82339  Sial, S., Sobolev gradient algorithm for minimum energy state of s-waves superconductors-finite element setting, Supercond sci. tecnol., 18, 675-677, (2005)  Nittka, R.; Sauter, M., Sobolev gradients for differential algebraic equations, Electron. J. differ. equat., 42, 1-31, (2008) · Zbl 1165.65373  Mujeeb, D.; Neuberger, J.W.; Sial, S., Recursive form of Sobolev gradient for ODEs on long interval, Int. J. comput. math., 87, 1727-1740, (2008) · Zbl 1154.65061  McQuarrie, D.A., Statistical mechanics, (1976), Harper & Row New York  FreeFem++ available at http://www.freefem.org/ff++.  Majid, A.; Sial, S., Application of Sobolev gradient to possion – boltzmann system, J. comput. phys., (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.