The Linear Finite Element Method for a Two-Dimensional Singular Boundary Value Problem

Abstract

The following model problem is studied:\[\Omega : - \left[ {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\beta \frac{{\partial u}}{{\partial r}}} \right) + \frac{\partial }{{\partial z}}\left( {\beta \frac{{\partial u}}{{\partial z}}} \right)} \right] = f,\]\[\Gamma _1 :u = 0\] where $\Omega $ is a bounded open domain with $r > 0$ in the $(r,z)$ plane, $\Gamma _1 = {{\partial \Omega } / {\Gamma _0 }}$, $\Gamma _0 = \partial \Omega \cap \{ (r,z):r = 0\} $. We introduce weighted Sobolev spaces $V^k (k = 1,2)$, and prove: (1) The problem has a unique solution u, and $u \in V_0^1 (\Omega ) \cap V^2 (\Omega )$. (2) The linear finite element solution $u_h $ exists and is unique. (3) The error $u - u_h $ in “energy norm” is of $O(h^2 )$. Particularly, if $\Omega $ is a polygon, then $\begin{gathered} \left\| {u - u_h } \right\|_{1,\Omega } = O(h) \hfill \\ \left\| {u - u_h } \right\|_{0,\Omega } = O\left( {h^2 } \right) \hfill \\ \end{gathered} $ where $\| \cdot \|_{k,\Omega } (k = 1,2)$ are the $V^k $ norms.