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


Author: Hengguang Li
Journal: Math. Comp. 78 (2009), 713-737
MSC (2000): Primary 35J70, 41A25, 41A50, 65N12, 65N15, 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-08-02179-0
Published electronically: September 2, 2008
MathSciNet review: 2476557
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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\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$.


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Additional Information

Hengguang Li
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Address at time of publication: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Email: li_h@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02179-0
Received by editor(s): October 18, 2006
Received by editor(s) in revised form: May 2, 2008
Published electronically: September 2, 2008
Additional Notes: H. Li was supported in part by NSF Grant DMS 0713743
Article copyright: © Copyright 2008 American Mathematical Society

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