Stability and accuracy of optimal local non-reflecting boundary conditions

Abstract

Problems in unbounded domains are often solved numerically by truncating the infinite domain via an artificial boundary B and applying some boundary condition on B , which is called a Non-Reflecting Boundary Condition (NRBC). Recently, a two-parameter hierarchy of optimal local NRBCs of increasing order has been developed. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition in the L 2 norm for functions in C ∞. The optimal NRBCs are combined with finite element discretization in the computational domain. Here the theoretical properties of the resulting class of schemes are examined. In particular, theorems are proved regarding the numerical stability of the schemes and their rates of convergence.