Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis

We develop and analyze Eulerian-Lagrangian localized adjoint methods (ellam) for convection-diffusion problems. The formulation uses space-time elements, with edges oriented along Lagrangian flow paths, in a time-marching scheme, where space-time test functions are chosen to satisfy a local adjoint condition. This allows Eulerian-Lagrangian concepts to be applied in a systematic mass-conservative manner to problems with general boundary conditions. In one space dimension with constant velocity, all combinations of inflow and outflow Dirichlet, Neumann, or flux boundary conditions are carefully considered, compared and discussed based on both analysis and numerical experiments. In some cases, the discrete unknowns include influxes, outfluxes, or resolution of the outflowing solution finer than the time-step size. Optimal-order error estimates in all cases and some superconvergence results are obtained. Numerical results show the strong potential of these methods and verify the theoretical estimates. Implementations for variable-coefficient problems in one and multiple space dimensions, considered in detail elsewhere, are outlined.