In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.

翻译：本文针对泊松方程的超弱变分形式，推导了唯一延拓问题与柯西问题的条件稳定性估计。数值近似通过正则化最小二乘泛函的极小化获得。其中出现的对偶范数被离散对偶范数替代，从而形成关于试验空间与检验空间的混合变分形式。对于稳定的空间对及正则化参数的适当选择，所得数值近似在子域上的$L_2$误差可由数据误差与最佳逼近误差之和的最佳分数次幂界定。与标准变分形式相比，后两种误差在更弱的范数下度量。为避免使用$C^1$有限元检验空间，亦可采用非协调有限元检验空间。此类空间或可导出定性相同的误差界，或在简化版本中导出包含附加数据振荡项的误差界。数值结果验证了理论结论。