Convergence of an Explicit Upwind Finite Element Method to Multi-Dimensional Conservation Laws

Abstract

An explicit upwind finite element method is given for the numerical computation to multi-dimensional scalar conservation laws. It is proved that this scheme is consistent to the equation and monotone, and the approximate solution satisfies discrete entropy inequality. To guarantee the limit of approximate solutions to be a measure valued solution, we prove an energy estimate. Then the $L^p$ strong convergence of this scheme is proved.