Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

In this work, the error of a given output functional is represented using bilinear forms that are different from those given by the adjoint problem. These representations can be employed to design novel h, p, and hp energy-norm and goal-oriented adaptive algorithms. Numerical results in 1D show that, for wave propagation problems, the advantages of this new representation are notorious when selecting the Laplace equation as the dual problem. Specifically, the computed upper bounds of the new error representation are sharper than the classical ones used in both energy-norm and goal-oriented adaptive methods, especially when the dispersion (pollution) error is significant.