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：考虑具有条件稳定性的不适定椭圆问题的有限元逼近。定义了包括网格参数收敛性和数据扰动在内的*最优误差估计*概念。收敛速率由底层连续问题的条件稳定性和有限元逼近空间的多项式阶数共同决定。证明表明，任何有限元逼近的收敛速率均无法超越该定义所确定的速率，从而验证了这一概念的有效性。回顾了近期提出的一类具有弱一致正则化的有限元方法，并证明其相关误差估计在我们定义的意义下是拟最优的。