Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem
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By: Christoph Lehrenfeld, Tim van Beeck, Igor Voulis
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 pollute... more
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.
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