Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law

摘要

A finite element method for a time dependent nonlinear hyperbolic conservation law in one space dimension (Burger's equation) is considered. The finite element method is higher order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. Assuming uniform boundedness of the finite element solutions, it is proved, using a compensated compactness result by Murat-Tartar, convergence to the entropy solution of the conservation law as the mesh parameter tends to zero.