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\begin{document}

\title{An exactly mass conserving and pointwise divergence free velocity
method: application to compositional buoyancy driven flow problems in
geodynamics}



\author[1]{Nathan Sime}%
\author[2]{Jakob M Maljaars}%
\author[1]{Cian R. Wilson}%
\author[1]{Peter van Keken}%
\affil[1]{Carnegie Institution for Science}%
\affil[2]{Delft University of Technology}%


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  \date{May 24, 2024}


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\begin{abstract}
Tracer methods are widespread in computational geodynamics for modeling
the advection of chemical data. However, they present certain numerical
challenges, especially when used over long periods of simulation time.
We address two of these in this work: the necessity for mass
conservation of chemical composition fields and the need for the
velocity field to be pointwise divergence free to avoid gaps in tracer
coverage. We do this by implementing the hybrid discontinuous Galerkin
(HDG) finite element method combined with a mass-conserving constrained
projection of the tracer data. To demonstrate the efficacy of this
system we compare it to other common finite element formulations of the
Stokes system and projections of the chemical composition. We provide a
reference of the numerical properties and error convergence rates which
should be observed by using these various discretization schemes. This
serves as a tool for verification of existing or new implementations. We
summarize these data in a reproduction of a published Rayleigh--Taylor
instability benchmark, demonstrating the importance of careful choices
of appropriate and compatible discretization methods for all aspects of
geodynamics simulations.%
\end{abstract}%



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