Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

Abstract:

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in $\mathbb {R}^{N}$. In a sense to be discussed below these sharpen known quasi–optimal $L_{\infty }$ and $W^{1}_{\infty }$ estimates for the error on irregular quasi–uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution $u$. We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in $\mathbb {R}^{N}$ and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non–smooth problems.