  We report \textit{ab initio} (LDA + U$_{sc}$) calculations of thermoelastic
properties of ferric iron (Fe$^{3+}$)- and aluminum (Al)-bearing bridgmanite
(MgSiO$_3$ perovskite), the main Earth forming phase, at relevant pressure and
temperature conditions and compositions. Three coupled substitutions, namely,
[Al]$_{Mg}$-[Al]$_{Si}$, [Fe$^{3+}$]$_{Mg}$-[Fe$^{3+}$]$_{Si}$, and
[Fe$^{3+}$]$_{Mg}$-[Al]$_{Si}$ have been investigated. Aggregate elastic moduli
and sound velocities are successfully compared with limited experimental data
available. In the case of the [Fe$^{3+}$]$_{Mg}$-[Fe$^{3+}$]$_{Si}$
substitution, the high-spin (S=5/2) to low-spin (S=1/2) crossover in
[Fe$^{3+}$]$_{Si}$ induces a volume collapse and elastic anomalies across the
transition region. However, the associated anomalies should disappear in the
presence of aluminum in the most favorable substitution, i.e.,
[Fe$^{3+}$]$_{Mg}$-[Al]$_{Si}$. Calculated elastic properties along a lower
mantle model geotherm suggest that the elastic behavior of bridgmanite with
simultaneous substitution of Fe$_{2}$O$_3$ and Al$_{2}$O$_3$ in equal
proportions or with Al$_{2}$O$_3$ in excess should be similar to that of
(Mg,Fe$^{2+}$)SiO$_3$ bridgmanite. Excess Fe$_{2}$O$_3$ should produce elastic
anomalies though.
