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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
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