Relaxation methods for gauge field equilibrium equations

This article gives a pedagogical introduction to relaxation methods for the numerical solution of elliptic partial differential equations, with particular emphasis on treating nonlinear problems with $\delta$-function source terms and axial symmetry, which arise in the context of effective Lagrangian approximations to the dynamics of quantized gauge fields. The authors present a detailed theoretical analysis of three models which are used as numerical examples: the classical Abelian Higgs model (illustrating charge screening), the semiclassical leading logarithm model (illustrating flux confinement within a free boundary or "bag"), and the axially symmetric Bogomol'nyi-Prasad-Sommerfield monopoles (illustrating the occurrence of topological quantum numbers in non-Abelian gauge fields). They then proceed to a self-contained introduction to the theory of relaxation methods and allied iterative numerical methods and to the practical aspects of their implementation, with attention to general issues which arise in the three examples. The authors conclude with a brief discussion of details of the numerical solution of the models, presenting sample numerical results.