  This paper considers weak Galerkin finite element approximations for a
quasistatic Maxwell viscoelastic model. The spatial discretization uses
piecewise polynomials of degree $k \ (k\geq 1)$ for the stress approximation,
degree $k+1$ for the velocity approximation, and degree $k$ for the numerical
trace of velocity on the inter-element boundaries. The temporal discretization
in the fully discrete method adopts a backward Euler difference scheme. We show
the existence and uniqueness of the semi-discrete and fully discrete solutions,
and derive optimal a priori error estimates. Numerical examples are provided to
support the theoretical analysis.
