The finite element method with nonuniform mesh sizes for unbounded domains

The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem: \[ − Δ u = f in Ω C , u = g on ∂ Ω , ∂ u ∂ r + 1 r u = o ( 1 r ) as r = | x | → ∞ , - Δ u = f {in}\;{Ω ^C}, u = g {on}\;∂ Ω , {{∂ u}}{{∂ r}} + 1/ru = o ( {1/r} ) {as}\;r = |x| → ∞ , \] where Ω C {Ω ^C} is the complement in R 3 {R^3} (three-dimensional Euclidean space) of a bounded set Ω Ω with smooth boundary ∂ Ω ∂ Ω , f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary