We consider an inverse problem where the forward problem is that of linear plane stress elasticity, or equivalently, that of linear heat/hydraulic conduction. We demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem. In particular, it is stable for the response variable and the Lagrange multiplier, but not for the material property field. This lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem. We overcome this difficulty by adding to the saddle point problem a residual-based term that provides sufficient stability, and prove optimal convergence in an energy-like norm. We verify these estimates through simple numerical examples. We note that while we have considered a specific model for an inverse elliptic problem in this manuscript, similar ideas could be developed for a broad class of inverse elliptic problems.

我们考虑一个反问题，其中正向问题是线性平面应力弹性的问题，或者等效地是线性热/液压传导的问题。我们证明了从最小化问题获得的鞍点问题的线性化版本继承了正向椭圆问题的稳定性。特别是对于响应变量和拉格朗日乘数是稳定的，但对于材料属性字段则不稳定。缺乏稳定性意味着我们无法通过网格细化证明整个问题的最优收敛性。我们通过向鞍点问题添加基于残差的项来提供足够的稳定性，从而克服了这一困难，并证明了类似能量范数的最佳收敛。我们通过简单的数字示例来验证这些估计。