Friedrichsand Poincaré-type inequalities are important and widely used in the area of partial differential equations and numerical analysis. Most of their proofs appearing in references are the argument of reduction to absurdity. In this paper, we give direct proofs of Friedrichs-type inequalities in H 1(Ω) and Poincaré-type inequalities in some subspaces of W1,p(Ω). The dependencies of the ine...

The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems first order partial differential equations (Friedrichs, 1958), nowadays better known as systems. Recently, Antoni\'c, Michelangeli Erceg (2017) presented purely operator-theoretic description allowing application the universal ope...

Four explicit finite difference schemes, including Lax-Friedrichs, Nessyahu-Tadmor, Lax-Wendroff and Lax-Wendroff with a nonlinear filter are applied to solve water hammer equations. The schemes solve the equations in a reservoir-pipe-valve with an instantaneous and gradual closure of the valve boundary. The computational results are compared with those of the method of characteristics (MOC), a...