Pointwise Error Estimates for Differences in Piecewise Linear Finite Element Approximations
A. H. Schatz, Lars B. Wahlbin · SIAM Journal on Numerical Analysis · 2003
We consider piecewise linear finite element approximations u h to u the solution of an elliptic boundary value problem. New estimates for the differences |e(x 1 )-e(x 2 )| (where e(x) = u(x)-u_h(x) is the error and x 1 and x 2 are any two points in the domain) are obtained in terms of weighted $L_\infty$ norms. As a consequence, so-called asymptotic expansion inequalities are derived that have been applied to obtain asymptotically exact a posteriori estimators for the gradient on each element.