Optimal Error Properties of Finite Element Methods for Second Order Elliptic Dirichlet Problems

We use the informational approach of Traub and Wozniakowski [9] to study the variational form of the second order elliptic Dirichlet problem Lu = f on ü C RN. For /e Hr(Q), where r> -1, a quasi-uniform finite element method using n linear functional Jaf^i nas T7'(ß)-norm error 0(n~ N/2. We show that when N = 1, there is no method using n function evaluations whose error is better than ñ(n~r); thus for N = 1, the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of /.