In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. Tikhonov regularisation is necessary for high order polynomial approximation, , but its weak consistency may be tuned to allow for optimal convergence. For low order elements a simple numerical scheme with optimal convergence is obtained without stabilization. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability and consistency assumptions. Numerical experiments are provided to illustrate the theoretical results.

翻译：本文针对泊松方程不适定问题的拟可逆解，发展了一种混合变分形式的离散化框架。通过精心选取试验空间与试探空间，我们得到了一种在特定残差范数下稳定的变分形式。数值稳定性与最优收敛性基于该问题的条件稳定性性质得以建立。对于高阶多项式逼近，Tikhonov正则化是必要的，但其弱一致性可通过调整以实现最优收敛。对于低阶单元，无需稳定化即可获得具有最优收敛性的简单数值格式。我们还为满足适当稳定性与一致性假设的有限元空间可行配对提供了指导准则。数值实验验证了理论结果。