The optimal convergence of finite element methods for the three-dimensional second-order elliptic boundary value problem with singularity

Abstract

Let x0∈Ω. Assume that Gx0(x) satisfies the elliptic boundary value problem LGx0(x)≡∂∂xj(aij(x)∂Gx0(x)∂xi)=δ(x−x0),inΩ,Gx0(x)=0on∂Ω, where Ω⊂ℜ3 is a convex polyhedral domain. In this article, we propose specially graded meshes Th. Furthermore, we obtain an L2(Ω)-error estimate of order hk+1|ln⁡h|32 for approximations with a Pk finite element method.