Goal-oriented adaptive finite element methods with optimal computational complexity

Abstract

We consider a linear symmetric and elliptic partial differential equation (PDE) and a linear goal functional. We design a goal-oriented adaptive finite element method (GOAFEM), which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method (PCG). We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates with respect to the number of degrees of freedom as well as the computational cost.