Error estimates of finite element method for a singular elliptic boundary-value problem (Q2762977)

The authors consider the Dirichlet problem for a \(2l\)th order elliptic partial differential operator defined in a non-smooth bounded domain \( \Omega \) verifying the cone property. The weak solution \( u \) of the problem under investigation admits a singularity at a conical point \( x^{0} \in \partial \Omega \), is unique and exists in the weighted Sobolev space \( W^{l,2}(\Omega, \mu) \) with weight function \( |x - x_{0}|^{\mu} \). The authors estimate the error \( u - u_{h} \), where \( u_{h} \) is the finite element solution of degree \( k \) in the sense of Galerkin, in an appropriate weighted norm and derive an optimal order of convergence \( O(h^{k+1-l}) \).