First-order approximation

Abstract

This chapter focuses on the approximation of nonlinear hyperbolic systems using finite elements. We describe a somewhat loose adaptation to finite elements of a scheme introduced by Lax. The method, introduced by Guermond, Nazarov, and Popov, can be informally shown to be first-order accurate in time and space and to preserve every invariant set of the hyperbolic system. The time discretization is based on the forward Euler method and the space discretization employs finite elements. The theory applies regardless of whether $H^1$-conforming or discontinuous elements are used.