Goal-Oriented Adaptivity using Unconventional Error Representations
In Goal-Oriented Adaptivity (GOA), the error in a Quantity of Interest (QoI) is represented using global error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient GOA. These representations can be employed to design novel h, p, and hp energy-norm and goal-oriented adaptive algorithms. While the method can be applied to a variety of problems, in this Dissertation we first focus on one-dimensional (1D) problems, including Helmholtz and steady-state convection-dominated diffusion problems. Numerical results in 1D show that for the Helmholtz problem, it is advantageous to select the Laplace operator for the alternative error representation. Specifically, the 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. The 1D steady-state convection-dominated diffusion problem with homogeneous Dirichlet boundary conditions exhibits a boundary layer that produces a loss of numerical stability. The new error representation based on the Laplace operator delivers sharper error upper bounds. When applied to a p-GOA, the alternative error representation captures earlier the boundary layer, despite the existing spurious numerical oscillations. We then focus on the two- and three-dimensional (2D and 3D) Helmholtz equation. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones. When using the alternative error indicators, a naive p-adaptive process converges, whereas under the same conditions, the classical method fails and requires the use of the so-called Projection Based Interpolation (PBI) operator or some other technique to regain convergence. We also provide guidelines for finding operators delivering sharp error representation upper bounds.