Goal-oriented adaptive finite element methods for elliptic problems revisited

A goal-oriented a posteriori error estimation of an output functional for elliptic problems is presented. Continuous finite element approximations are used in quadrilateral and triangular meshes. The algorithm is similar to the classical dual-weighted error estimation, however the dual weight contains solutions of the proposed patch problems. The patch problems are introduced to apply Clément and Scott–Zhang type interpolation operators to estimate point values with the finite element polynomials. The algorithm is shown to be reliable, efficient and convergent.