The main purpose of this work is to study continuous finite element methods for hyperbolic problems. In scalar case, it is shown that using consistent mass matrix is not compatible with the maximum principle. Moreover, we propose two algorithms which preserve the maximum principle and have high order convergence at the same time. For hyperbolic systems, such as Euler equations, we propose two methods which keep the invariant domain property even in Arbitrary Lagrangian Eulerian (ALE) framework.
