Multi-goal-oriented anisotropic error control and mesh adaptivity for time-dependent convection-dominated problems
We present an anisotropic multi-goal error control strategy for time-dependent convection-diffusion-reaction (CDR) equations based on the Dual Weighted Residual (DWR) method. The approach enables accurate and efficient control of discretization errors with respect to multiple goal functionals simultaneously, a crucial aspect in many real-world applications.
The use of finite elements with anisotropic polynomial degree allows for elementwise separation of error contributions, both temporally and spatially, and further decomposes spatial contributions directionally. This yields highly localized, directionally resolved error indicators that naturally guide anisotropic mesh refinement strategies.
The resulting adaptive meshes efficiently capture sharp layers and directional features of the solution aligned with the goals. Numerical experiments on established benchmark problems for convection-dominated transport demonstrate the efficiency, accuracy, and robustness of the proposed approach compared to isotropic refinement and uniform mesh strategies.
In addition to numerical results, we also discuss the theoretical justification of the anisotropic DWR approach and present implementation aspects of the required anisotropic finite element spaces.
We give a brief outlook on how the framework can be extended to include fully discontinuous discretizations in space and time. In that case, the anisotropic error splitting can be used to steer directional hp-refinement.
