Residual distribution finite element method for convection-dominated problems

In this paper, we present a new stabilized finite element method for the simulation of nonlinear convection-dominated systems of PDEs. We seek to combine the robustness of upwind finite-volume methods for the discretization of strongly shocked flows with the ease and accuracy of C0C0 finite element reconstruction. This is achieved by exploiting the similarities between traditional SUPG method and finite volume residual distribution schemes. By constructing the finite element analog of finite-volume fluctuation-splitting schemes, we combine the ability to simulate PDEs with strongly discontinuous solutions of upwind finite-volume methods with the reliable accuracy of finite element reconstruction, which traditional finite-volume methods often lack on unstructured grids. Finally, we test the proposed algorithm with challenging problems drawn from hypersonic compressible flow and semiconductor device simulation.