While exponential face bubble functions have been shown to provide stability and accuracy in the discretization of reaction-dominated diffusion problems, they may have limitations in terms of computational efficiency and implementation complexity. One potential drawback is the computational cost associated with evaluating and storing the exponential functions, especially for problems with a large number of elements or high-dimensional domains.
Another limitation could be the sensitivity of the method to the choice of parameters, such as the decay rate of the exponential functions. If not chosen appropriately, it could lead to numerical instabilities or inaccuracies in the solution.
Alternative approaches that could be explored include using polynomial or piecewise polynomial basis functions with tailored properties to capture the boundary and interior behavior of the solution. These functions could be designed to have compact support or specific decay properties to ensure stability and efficiency in the numerical scheme.
Furthermore, exploring adaptive strategies to dynamically adjust the shape and properties of the basis functions based on the local solution behavior could also be a promising direction to address the limitations of exponential face bubble functions.