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A fully coupled high order discontinuous Galerkin (DG) solver for viscoelastic Oldroyd B fluid flow problems is presented. Contrary to known methods combining DG for the discretization of the convective terms of the material model with standard finite element methods (FEM) and using elastic viscous stress splitting (EVSS) and its derivatives, a local discontinuous Galerkin (LDG) formulation first described for hyperbolic convection‐diffusion problems is used. The overall scheme is described, including temporal and spatial discretization as well as solution strategies for the nonlinear system, based on incremental increase of the Weissenberg number. The solvers suitability is demonstrated for the two‐dimensional confined cylinder benchmark problem. The cylinder is immersed in a narrow channel with a blocking ratio of 1:2 and the drag force of is compared to results from the literature. Furthermore, steady and unsteady calculations give a brief insight into the characteristics of instabilities due to boundary layer phenomena caused by viscoelasticity arising in the narrowing between channel and cylinder.