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

Abstract:

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 $\Omega$, $u|_{\partial \Omega }=0$ 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., $||u-u_n||_{H^1(\Omega )}\leq C\textrm {{dim}}(V_n)^{-\frac {m}{2}}||f||_{H^{m-1}(\Omega )}$ with $C$ independent of $f$ and $n$.