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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 Q C RN. For f G HW(Q), where r -1, a quasi-uniform finite element method using n linear functionals fs f4, has H'(Q)-norm error J(n (r+ 1)/N). We prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals. An analogous result holds if L is of order 2m: if f G Hr(Q), where r -m, then there is a finite element method whose Ha(Q)-norm error is n( (2m+r-a)/N) for 0 N/2. We show that when N = 1, there is no method using n function evaluations whose error is better than Q(n r); thus for N = 1, the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f.

Optimal error properties of finite element methods for second order elliptic Dirichlet problems