On the use of adjoint-based sensitivity estimates to control local mesh refinement

The goal of efficient and robust error control, through local mesh adaptation in the computational solution of partial differential equations, is predicated on the ability to identify in an a posteriori way those localized regions whose refinement will lead to the most significant reductions in the error. The development of a posteriori error estimation schemes and of a refinement infrastructure both facilitate this goal, however they are incomplete in the sense that they do not provide an answer as to where the maximal impact of refinement may be gained or what type of refinement - elemental partitioning (h-refinement) or polynomial enrichment (p-refinement) - will best lead to that gain. In essence, one also requires knowledge of the sensitivity of the error to both the location and the type of refinement. In this communication we propose the use of adjoint-based sensitivity analysis to discriminate both where and how to refine. We present both an adjoint-based and an algebraic perspective on defining and using sensitivities, and then demonstrate through several one-dimensional model problem experiments the feasibility and benefits of our approach.