Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation Original Research Article

Eulerian-Lagrangian localized adjoint methods (ELLAM) are developed for the nonlinear Buckley-Leverett equation, which is characterized by degenerate diffusion and sharpening near-shock solutions. The ELLAM methodology employs space-time finite elements with edges oriented along flow paths, and space-time test functions that satisfy a local adjoint condition. This combination extends Eulearian-Lagrangian concepts in a systematic mass-conservative fashion to problems with general boundary conditions. Various kinds of boundary conditions are considered, and a local time-stepping procedure is developed for a no-flow outlet condition that leads to a boundary layer. Numerical experiments illustrate the potential of these methods.