Sobolev gradient preconditioning for the electrostatic potential equation

Energy minimization related to the nonlinear Schrödinger equation

“…The recent theory of Sobolev gradients [1] provides a unified point of view on such problems, both in function spaces and in finite dimensional approximations to such problems. Sobolev gradients have been used for ODE problems [1,2] in a finite-difference setting, PDEs in finite-difference [2] and finite-element settings [3], minimizing energy functionals associated with Landau-Ginzburg models in finite-difference [4] and finite-element [5,6] settings, the electrostatic potential equation [7], nonlinear elliptic problems [8], semilinear elliptic systems [9], simulation of Bose-Einstein condensates [10], and inverse problems in elasticity [11] and groundwater modeling [12]. The Sobolev gradient technique has been discussed previously for minimizing Schrödinger functionals in Fourier space setting [13].…”

Numerical Solutions of Singularly Perturbed Reaction Diffusion Equation with Sobolev Gradients

“…In Sobolev gradient methods, linear operators are formed to improve the condition number in the steepest descent minimization process. The efficiency of Sobolev gradient methods has been shown in many situations, for example, in physics [4][5][6][7][8][9][10][11], image processing [12,13], geometric modelling [14], material sciences [15][16][17][18][19][20], Differential Algebraic Equations, (DAEs) [21] and for the solution of integrodifferential equations [22].…”

Approximating time evolution related to Ginzburg–Landau functionals via Sobolev gradient methods in a finite-element setting

“…The recent theory of Sobolev gradients [2] gives a unified approach for such problems. Sobolev gradients have been used for ODEs [2,3] in a finite-difference setting, PDEs in finite-difference [3] and finite-element settings [4], minimizing energy functionals associated with Ginzburg-Landau models in finite-difference [5] and finite-element [6,7] settings, the electrostatic potential equation [8], nonlinear elliptic problems [9], semilinear elliptic systems [10], simulation of Bose-Einstein condensates [11], inverse problems in elasticity [12] and groundwater modelling [13]. [2] has a detailed analysis regarding the construction and the application of Sobolev gradients.…”