We consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation 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 approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.

我们考虑具有条件稳定性的不适定椭圆问题的数值近似。最优误差估计的概念被定义为包括离散化的收敛性和数据的扰动。收敛速度由基础连续问题的条件稳定性和逼近空间的多项式阶数决定。证明了在不增加对扰动的敏感性的情况下，没有任何近似可以以比定义给出的速率更好的速率收敛，从而证明了该概念的合理性。回顾最近引入的一类具有弱一致正则化的原始对偶有限元方法，并且相关的误差估计在该定义的意义上被证明是最优的。