A-priori analysis and the finite element method for a class of degenerate elliptic equations

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

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