  We introduce a model which integrates the complex Ginzburg-Landau (CGL)
equation in two dimensions (2D) with the linear-cubic-quintic combination of
loss and gain terms, self-defocusing nonlinearity, and a periodic potential. In
this system, stable 2D \textit{dissipative gap solitons} (DGSs) are
constructed, both fundamental and vortical ones. The soliton families belong to
the first finite bandgap of the system's linear spectrum. The solutions are
obtained in a numerical form and also by means of an analytical approximation,
which combines the variational description of the shape of the fundamental and
vortical solitons and the balance equation for their total power. The
analytical results agree with numerical findings. The model may be implemented
as a laser medium in a bulk self-defocusing optical waveguide equipped with a
transverse 2D grating, the predicted DGSs representing spatial solitons in this
setting.
