  An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is
studied in this paper, in addition to its convergence and optimality. The
modified weak Galerkin bilinear form is simplified without the need of the
skeletal variable, and the approximation space is chosen as the discontinuous
polynomial space as in the discontinuous Galerkin method. Upon a reliable
residual-based a posteriori error estimator, an adaptive algorithm is proposed
together with its convergence and quasi-optimality proved for the lowest order
case. The primary tool is to bridge the connection between the modified weak
Galerkin method and the Crouzeix-Raviart nonconforming finite element. Unlike
the traditional convergence analysis for methods with a discontinuous
polynomial approximation space, the convergence of AmWG is penalty parameter
free. Numerical results are presented to support the theoretical results.
