A Space-Time Adaptive Finite Element Algorithm Based on Dual Weighted Residual Methodology for Parabolic Equations

We propose in this paper a space-time adaptive algorithm based on the dual weighted residual (DWR) idea in the framework of a finite element method. Our algorithm consists of applying the DWR technique locally in each time interval $I_n:=(t_{n-1},t_n]$, and thus, we control the local or truncation error for a functional of the solution $J(u)$. The disadvantage of our method compared to the traditional DWR method is that having a good local error control does not guarantee that the global error will be bounded. However, the main advantage of our method is that we avoid solving backward in the whole time interval $I:=(0,T]$ the associated linear dual problem so that the storage requirements are smaller. This means that we can define a self-sufficient criterion that allows us to have control of the time step $\Delta t$ and the mesh size h as time of integration progresses. Another good feature of our algorithm is the extension of the spatial postprocessing procedure of the traditional DWR method to unstructu...