RESUMEN

In this paper, the technique of construction and analysis of quasimonotone finite-difference numerical schemes for scalar conservation laws in one space dimension, developed in Part I [SIAM J. Numer. Anal., 26 (1989), pp. 1325–1341], is extended to a wide class of Petrov–Galerkin finite-element methods. The resulting schemes are called the quasimonotone finite-element schemes. The approximate solution u_h is written as ū_h  + ŭ_h , where ū_h is a piecewise-constant function. The Petrov–Galerkin methods are then considered to be a set of equations that defines “the parameter” ŭ_h, plus a single equation, which is essentially a finite-difference scheme, that defines “the means” ŭ_h. All the results of the theory of quasimonotone finite-difference schemes can be coupling.