Develop derivative-jump spatial smoothness indicators for parabolic equations

Develop and validate spatial smoothness indicators based on derivative jumps for parabolic equations, including the discrete Green’s-function constructions needed to connect the continuous and semi-discrete solutions in local spatial error estimation.

Background

The paper develops spatial smoothness indicators for piecewise-polynomial approximations by scaling jumps of derivatives across cell interfaces. These indicators are shown to be necessary for optimal-order approximation of smooth functions in several norms.

The notes state that this derivative-jump framework has not yet been successfully transferred to parabolic equations. They suggest that discrete Green’s functions may be required to relate the exact evolution solution to the semi-discrete solution during local spatial error analysis; existing work is described as limited to one-dimensional linear parabolic equations.

References

For parabolic equations, the spatial smoothness indicators based on the derivative jumps has bot been successfully used to parabolic equations. This may involve some form of discrete Green's functions to build bridges between $\tilde{u}(t)$ and $uh(t)$ in local spatial error estimation.

— Numerical Smoothness and Error Analysis for Evolution Equations  (2610.12888 - Sun, 9 Oct 2026) in Section 4, “Future research”