Generalized Green's Functions and the Effective Domain of Influence

One well-known approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized Green’s function. The generalized Green’s function solves the adjoint problem with data related to a quantity of interest and measures the effects of stability, including any decay of influence characteristic of elliptic problems. We show that consideration of the generalized Green’s function can be used to improve the efficiency of the solution process when the goal is to compute multiple quantities of interest and/or to compute quantities of interest that involve globally supported information such as average values and norms. In the latter case, we introduce a solution decomposition in which we solve a set of problems involving localized information and then recover the desired information by combining the local solutions. By treating each computation of a quantity of interest independently, the maximum number of elements required to achieve the desired accuracy can be decreased significantly.