The Finite Element Method for a Class of Degenerate Elliptic Equations

Consider the degenerate elliptic operator Lδ := −∂2 x − δ x2 ∂ 2 y on Ω := (0, 1) × (0, l), for δ > 0, l > 0. We prove well-posedness and regularity results for the second-order degenerate elliptic equation Lδu = f in Ω, u|∂Ω = 0 using weighted Sobolev spaces Km a . In particular, by a proper choice of the parameters in the weighted Sobolev spaces Km a , we establish the existence and uniqueness of the solution. In addition, we show that there is no loss of Km a -regularity for the solution of the equation. We then provide an explicit construction of a sequence of finite dimensional subspaces Vn for the finite element method, such that the optimal convergence rate is attained for un ∈ Vn, i.e., ||u − un||H1(Ω) ≤ Cdim(Vn) −2 ||f ||Hm−1(Ω) with C independent of f and n.