  Perovskite structure SrMnO$_3$ is a rare example of a multiferroic material
where strain-tuning and/or cation substitution could lead to coinciding
magnetic and ferroelectric ordering temperatures, which would then promise
strong magnetoelectric coupling effects. Here, we establish the temperature and
strain dependent ferroic phase diagram of SrMnO$_3$ using
first-principles-based effective Hamiltonians. All parameters of these
Hamiltonians are calculated using density functional theory, i.e., no fitting
to experimental data is required. Temperature dependent properties are then
obtained from Monte Carlo and molecular dynamics simulations. We observe a
sequence of several magnetic transitions under increasing tensile strain, with
a moderate variation of the corresponding critical temperatures. In contrast,
the ferroelectric Curie temperature increases strongly after its onset around
2.5\,\% strain, and indeed crosses the magnetic transition temperature just
above 3\,\% strain. Our results indicate pronounced magnetoelectric coupling,
manifested in dramatic changes of the magnetic ordering temperatures and
different magnetic ground states as function of the ferroelectric distortion.
In addition, coexisting ferroelectric and ferromagnetic order is obtained for
strains above 4\,\%. Our calculated phase diagram suggests the possibility to
control the magnetic properties of SrMnO$_3$ through an applied electric field,
significantly altering the magnetic transition temperatures, or even inducing
transitions between different magnetic states.
