Title:

Generalized Green's Functions and the
Effective Domain of Influence

Abstract:

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.