Optimal control using flux potentials: A way to construct bound-preserving finite element schemes for conservation laws

To ensure preservation of local or global bounds for numerical solutions conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty the proposed methodology lies in use potentials as control variables and targets inequality-constrained optimization problems fluxes. In contrast to optimal via general source terms, discrete property flux-corrected approximations is guaranteed without need impose additional equality constraints. Since number less than fluxes multidimensional case, potential-based version involves fewer unknowns direct calculation We show that feasible set potential-state potential-target (PP) problem nonempty choose primal–dual Newton method calculating potentials. results studies linear nonlinear laws 2D demonstrate superiority new OB-PP algorithms closed-form limiting under worst-case assumptions.