  A graph is $2$-planar if it has local crossing number two, that is, it can be
drawn in the plane such that every edge has at most two crossings. A graph is
maximal $2$-planar if no edge can be added such that the resulting graph
remains $2$-planar. A $2$-planar graph on $n$ vertices has at most $5n-10$
edges, and some (maximal) $2$-planar graphs -- referred to as optimal
$2$-planar -- achieve this bound. However, in strong contrast to maximal planar
graphs, a maximal $2$-planar graph may have fewer than the maximum possible
number of edges. In this paper, we determine the minimum edge density of
maximal $2$-planar graphs by proving that every maximal $2$-planar graph on
$n\ge 5$ vertices has at least $2n$ edges. We also show that this bound is
tight, up to an additive constant. The lower bound is based on an analysis of
the degree distribution in specific classes of drawings of the graph. The upper
bound construction is verified by carefully exploring the space of admissible
drawings using computer support.
