Maximum principle preserving continuous finite element schemes for conservation laws

Murtazo Nazarov
Division of Scientific Computing
Department of Information Technology
Uppsala University


The question of stabilization of Galerkin finite element methods for conservation laws is not yet fully resolved. Once the stabilization terms are added to the system the new questions typically arise, for example on the parameters and meshsizes, computational costs, convergence to the right physical solution, etc. In this talk we will introduce a new stabilized continuous Lagrange finite element method for the explicit approximation of scalar conservation equations that does not require any a priori knowledge of quantities like local wave-speed, proportionality constant, or local mesh-size. Provided the lumped mass matrix is positive definite and the flux is Lipschitz, the method is proved to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions without any particular regularity assumption. Then, using the Boris-Book-Zalesak flux correction technique, we extend the method to (at least) a second-order accuracy that satisfy maximum principle. The method is tested on a series of linear and nonlinear benchmark problems.