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

Consider the degenerate elliptic operator $\mathcal{L_\delta} := -\partial^2_x-\frac{\delta^2}{x^2}\partial^2_y$ on $\Omega:= (0, 1)\times(0, l)$, for $\delta>0, l>0$. We prove well-posedness and regularity results for the degenerate elliptic equation $\mathcal{L_\delta} u=f$ in $\Omega$, $u\vert _{\partial\Omega}=0$ using weighted Sobolev spaces $\mathcal{K}^m_a$. In particular, by a proper choice of the parameters in the weighted Sobolev spaces $\mathcal{K}^m_a$, we establish the existence and uniqueness of the solution. In addition, we show that there is no loss of $\mathcal{K}^m_a$-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 $u_n\in V_n$, i.e., $\vert\vert u-u_n\vert\vert _{H^1(\Omega)}\leq C{\rm {dim}}(V_n)^{-\frac{m}{2}}\vert\vert f\vert\vert _{H^{m-1}(\Omega)}$ with $C$ independent of $f$ and $n$.