A-priori analysis and the finite element method for a class of degenerate elliptic equations. (English) Zbl 1198.35111
Summary: 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., \( || u-u_n|| _{H^1(\Omega)}\leq C{\dim}(V_n)^{-\frac{m}{2}}|| f|| _{H^{m-1}(\Omega)}\) with \( C\) independent of \( f\) and \( n\).
MSC:
| 35J70 | Degenerate elliptic equations |
| 41A25 | Rate of convergence, degree of approximation |
| 41A50 | Best approximation, Chebyshev systems |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
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