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

We develop and analyze Eulerian-Lagrangian localized adjointmethods (ellam) for convection-diffusion problems. The formulationuses space-time elements, with edges oriented along Lagrangianflow paths, in a time-marching scheme, where space-time testfunctions are chosen to satisfy a local adjoint condition. Thisallows Eulerian-Lagrangian concepts to be applied in a systematicmass-conservative manner to problems with general boundary conditions.In one space dimension with constant velocity, all combinationsof inflow and outflow Dirichlet, Neumann, or flux boundary conditionsare carefully considered, compared and discussed based on bothanalysis and numerical experiments. In some cases, the discreteunknowns include influxes, outfluxes, or resolution of the outflowingsolution finer than the time-step size. Optimal-order errorestimates in all cases and some superconvergence results areobtained. Numerical results show the strong potential of thesemethods and verify the theoretical estimates. Implementationsfor variable-coefficient problems in one and multiple spacedimensions, considered in detail elsewhere, are outlined.