The filling process in injection molding involves polymer melt and air with large density/viscosity ratios and the transient free surface. It is treated as a challenging viscoelastic-Newtonian two-phase flow problem especially on irregular domains. In this paper, the complex filling process for the irregular cavity is simulated using a combined finite element/discontinuous Galerkin/level set method with application to the socket with five inserts, which is rarely investigated. The rheological behaviour of the viscoelastic fluid is predicted according to the eXtended Pom-Pom (XPP) constitutive model. The level set method proposed from prior literature is utilized to capture the moving interface because of its simplicity and efficiency. The hybrid continuous and discontinuous Galerkin method is utilized to solve the viscoelastic incompressible Navier-Stokes equations. The second order Runge Kutta discontinuous Galerkin (RKDG) method is employed to deal with the XPP constitutive equation and the level set equation due to their hyperbolic natures. This combined algorithm is convenient to cope with the irregular cavity and avoid any stabilization terms. We first investigate the filling process of the rectangular cavity without and with a diamond insert and compare with other numerical and experimental results to illustrate the validity of the coupled method. Moreover, the cavity of the socket with five inserts is considered an application case. We analyze the influences of the inlet velocity and elasticity on physical quantities such as stresses, stretch, etc. The simulation results could provide some numerical predictions for the polymer industry.