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

We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). 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. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function u, then u is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers' equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments. 0. Introduction. In this note we prove some results concerning the convergence of a higher-order accurate finite element method for a nonlinear hyperbolic conservation law. As far as we know, no earlier theoretical results for such methods are available. We shall consider Burgers' equation in one space dimension, i.e., the problem of finding a scalar function u(x, t) such that (0.1a) u, + uux = 0, x e R = (-00,00), t> 0, (0.1b) u(0,x) = u0(x), x 0, together with a certain modification of the test functions giving added stability. We shall first prove that if a sequence of finite element solutions converges (as the mesh size h tends to zero) boundedly almost everywhere to a function u, then u is an entropy solution of (0.1). Thus, streamline diffusion finite element solutions cannot converge to a weak solution not satisfying Received March 6, 1986; revised September 15, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 65M15.