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：本文提出了一种新的H(div)相容的非拟合有限元方法，用于求解混合泊松问题。该方法在切割构型下具有鲁棒性，并保持了体拟合有限元方法的守恒性质。关键在于将有源网格（而非物理域）上的散度约束进行公式化，从而在不引入污染质量平衡的稳定化项的前提下，获得对切割构型的鲁棒性。这一公式变化导致轻微的不一致性，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局部后处理方法对标量变量进行后处理，我们为两个变量都保留了最优收敛率，甚至在标量变量后处理后实现了超收敛。本文介绍了该方法，对其进行了严格的先验误差分析，并讨论了若干变体与扩展。数值实验验证了理论结果。