An optimal-order estimate for Eulerian-Lagrangian localized adjoint methods for variable-coefficient advection-reaction problems

Previously, we developed Eulerian-lagrangian localized adjoint methods (ELLAMs) to solve advection-reaction problems. ELLAMs provide a systematic approach to treat boundary conditions and to conserve mass and have proven to be powerful methods for advection-dominated problems. In this paper, the authors conduct theoretical analysis for these ELLAMs and prove that they have an optimal-order convergence rate. Moreover, the estimates involve only the derivatives of the exact solution in space and along the characteristics. In contrast, many existing methods have only suboptimal-order error estimates for advection-reaction problems and the estimates involve the temporal derivatives of the exact solution, which are usually much larger than the derivatives along the characteristics.