Duality-based adaptive refinement for elliptic pdes

In this work we consider second order linear elliptic partial differential equations in two spatial dimensions: -1˙a1u+ b˙ 1u+cu =fin W u=0 on6W1 a1u˙ n=b on6W2. In some applications it is not the solution u of the PDE which is of primary interest, but some functional G of it. In such instances, if ũ is a computed approximation of u, it makes more sense to consider the approximation error in terms of G directly, namely |G( u − ũ)|, than it does to consider it in terms of some norm. In this work we develop an algorithm for computing asymptotically exact a posteriori estimates of the error |G( u − ũ)|, and local error indicators for use in adaptive refinement. The algorithm is based on the solution of an auxiliary problem which is defined on the same computational mesh as ũ. Although there are many important physical applications of functionals of solutions to PDEs in areas ranging from fluid dynamics to structural mechanics, we focus on its application to the problem of determining the influence of global information on the local behavior of the solution and error. In particular, functionals are chosen which approximate various norms of the error in a given subdomain, and the solution of the related auxiliary problem gives indication as to which regions outside this subdomain influence the error there in a nonnegligible way. The primary motivation for this focus is the potential application for parallel adaptive algorithms. A key component in such algorithms is the decomposition of the given domain into several smaller subdomains—one for each available processor. Each processor is given the task of adequately resolving the portion of u corresponding to its subdomain, so it is natural to want to determine the effect that approximation errors outside the subdomain of interest have on error within the subdomain.