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

Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem