Convergence of discontinuous Galerkin schemes for the Euler equations via dissipative weak solutions

•Introduction of dissipative weak solutions for the Euler equations in the framework of high-order FE methods.•First investigation of convergence of high-order FE schemes in terms of dissipative weak solutions for the Euler equations.•Consistency analysis of high-order DG methods without any underlying smoothness assumptions.•Recipe of obtaining a convergence result in terms of dissipative weak solutions.•Numerical validation of convergence using Cesaro averages for the Kelvin–Helmholtz problem.