  In this paper we present an immersed weak Galerkin method for solving
second-order elliptic interface problems on polygonal meshes, where the meshes
do not need to be aligned with the interface. The discrete space consists of
constants on each edge and broken linear polynomials satisfying the interface
conditions in each element. For triangular meshes, such broken linear
plynomials coincide with the basis functions in immersed finite element methods
[26]. We establish some approximation properties of the broken linear
polynomials and the discrete weak gradient of a certain projection of the
solution on polygonal meshes. We then prove an optimal error estimate of our
scheme in the discrete $H^1$-seminorm under some assumptions on the exact
solution. Numerical experiments are provided to confirm our theoretical
analysis.
