In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.

翻译：本文提出一种新的H(div)相容非拟合有限元方法，用于求解混合泊松问题，该方法在切割网格构型下具有鲁棒性，并保持体拟合有限元方法的守恒特性。关键在于将散度约束定义在活动网格而非物理域上，从而无需引入污染质量平衡的稳定化项，即可获得对切割构型的鲁棒性。这一公式变化虽导致轻微非一致性，但不会影响通量变量的精度。通过借鉴体拟合方法中经典的标量变量局部后处理技术，我们保持了两种变量的最优收敛速率，甚至实现标量变量后处理后的超收敛性。本文给出了该方法的具体构造，进行了严格的先验误差分析，并探讨了多种变体与扩展形式。数值实验验证了理论分析结果。