Sobolev preconditioning for the Poisson–Boltzmann equation

Application of Sobolev gradient method to Poisson–Boltzmann system

“…The PBE is an elliptic partial differential equation that models many important phenomena such as charge distributions in semiconductor devices [2] and the protein-folding problem. The PBE is being extensively studied to analyze the properties of the biomolecules in physics and chemistry.…”

“…The idea of a weighted gradient, was introduced by Mahavier in [1] who demonstrated its effectiveness in dealing with linear and nonlinear singular differential equations. Sobolev gradients were considered as preconditioners for solution of first order and second order differential equations in [2] in which a one dimensional PBE that arises in semiconductor modeling is considered. An example is also given in [2] of a problem for which the Sobolev gradient method converges but Newton's method does not.…”

“…Sobolev gradients were considered as preconditioners for solution of first order and second order differential equations in [2] in which a one dimensional PBE that arises in semiconductor modeling is considered. An example is also given in [2] of a problem for which the Sobolev gradient method converges but Newton's method does not. In this article, we combine the idea of preconditioning with a weighted Sobolev gradient and present its application to linear and nonlinear PBE.…”

Direct Minimization of the least-squares spectral element functional – Part I: Direct solver

“…A variety of approaches is available, from Jacobi pre-conditioning in the Conjugate Gradient solver, [24,25] to Sobolev pre-conditioning, [17,18].…”

A note on least squares methods

“…For example, in Reference [28] the Sobolev gradient steepest descent approach with optimal step size, when applied to a simple 1-D linear ordinary di erential equation test problem, converges in 16 steps independent of mesh size, whereas descent with the standard gradient and 50 gridpoints requires 40 515 steps. Hence one may also view determination of this natural descent direction as a preconditioning operation [28][29][30]. Note, however that this Sobolev descent requires an additional solve calculation at each descent step.…”