We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition.

翻译：本文考虑具有条件稳定性的不适定椭圆问题的有限元逼近。定义了包含网格参数收敛性和数据摄动收敛性的“最优误差估计”概念。收敛速率由原连续问题的条件稳定性以及有限元逼近空间的多项式阶数共同决定。本文证明，没有任何有限元逼近能够以优于该定义所给出的收敛速率收敛，从而验证了这一概念的合理性。同时回顾了近期提出的一类具有弱一致正则化的有限元方法，并证明其关联误差估计在本文定义框架下具有拟最优性。