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.