Renormalization group and variational treatments for a d-dimensional sine-Gordon model with correlated random fields
We study the effects of correlated symmetry-breaking-like random fields on critical properties of a d-dimensional sine-Gordon model. A Wilsonian renormalization group analysis shows an unusual scenario in the absence of stable fixed points. The physical nature of the related runaway in the parameter space is investigated by means of a complementary variational treatment. Self-consistent predictions for d>1 suggest that the disorder of the type considered here inhibits any phase transition with diverging correlation length and that the system lives in a “glassy-phase” at all temperatures.
Given ?>0 andp?(0,1), we consider the following problem: findu such that $$\begin{gathered} - \Delta u + \lambda [u]_ + ^p = 0in\Omega , \hfill \\ u = 1on\partial \Omega , \hfill \\ \end{gathered} $$ whereΩ??2 is a smooth convex domain. We prove optimalH 1 andL ? error bounds for the standard continuous piecewise linear Galerkin finite element approximation. In addition we analyse a more practical approximation using numerical integration on the nonlinear term. Finally we consider a modified nonlinear SOR algorithm, which is shown to be globally convergent, for solving the algebraic system derived from the more practical approximation.