Adaptive evolution of the CARE approximation space

Develop a CARE spatial approximation space that adapts to the evolving solution, allowing the spatial representation to respond dynamically to changes in solution structure.

Background

The CARE-SAV framework constructs a fixed spatial approximation space before time integration by selecting numerically independent directions from a redundant random-feature dictionary. The paper establishes stability and convergence for this fixed-space formulation, but the solution-dependent spatial structures may change substantially during long-time evolution or in problems with moving interfaces and coarsening patterns.

The authors explicitly identify dynamic adaptation of the CARE space as an unresolved question beyond the present study. Such an extension would allow the basis to be updated in response to evolving solution features while raising the need to preserve energy stability, efficient solvability, and reliable error control under space changes.