The numerical analysis of higher-order nonlinear FE method for advection dominated problems

Abstract

The numerical analysis of nonlinear discontinuity-capturing methods applied to advection dominated problems has not been completely established yet. Some particular results give very good contribution towards the existence of discrete solutions although no uniqueness results are demonstrated. This paper studies the conditions for the uniqueness of the solution when using to the CAU (Consistent Approximate Upwind) Petrov-Galerkin for solving advection dominated problems. The main issue in this analysis is that it relies on the optimality property of the CAU solution which does not depend on any restrictions of the approximation spaces.