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-t ime elements, with edges oriented along Lagrangian flow paths, in a ti me-marching scheme, where space-time test functions are chosen to sati sfy a local adjoint condition. This allows Eulerian-Lagrangian concept s to be applied in a systematic mass-conservative manner to problems w ith general boundary conditions. In one space dimension with constant velocity, all combinations of inflow and outflow Dirichlet, Neumann, o r flux boundary conditions are carefully considered, compared and disc ussed based on both analysis and numerical experiments. In some cases, the discrete unknowns include influxes, outfluxes, or resolution of t he outflowing solution finer than the time-step size. Optimal-order er ror estimates in all cases and some superconvergence results are obtai ned. Numerical results show the strong potential of these methods and verify the theoretical estimates. Implementations for variable-coeffic ient problems in one and multiple space dimensions, considered in deta il elsewhere, are outlined.