Boundary pointwise error estimate for finite element method

This paper is devoted to the pointwise error estimate up to boundary for the standard finite element solution of Poisson equation with Dirichlet boundary condition. Our new approach uses the discrete maximum principle for the discrete harmonic solution. Once the mesh in our domain satisfies the β- condition defined by us, the discrete harmonic solution with Dirichlet boundary condition has the discrete maximum principle and the pointwise error should be bounded by L₁- errors newly obtained.