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In this paper we present an immersed weak Galerkin method for solvingsecond-order elliptic interface problems on polygonal meshes, where the meshesdo not need to be aligned with the interface. The discrete space consists ofconstants on each edge and broken linear polynomials satisfying the interfaceconditions in each element. For triangular meshes, such broken linearplynomials coincide with the basis functions in immersed finite element methods[26]. We establish some approximation properties of the broken linearpolynomials and the discrete weak gradient of a certain projection of thesolution on polygonal meshes. We then prove an optimal error estimate of ourscheme in the discrete $H^1$-seminorm under some assumptions on the exactsolution. Numerical experiments are provided to confirm our theoreticalanalysis.

An Immersed Weak Galerkin Method for Elliptic Interface Problems on Polygonal Meshes