High-order accuracy of the DPM with implicit geometry

Let the underlying discretization have second- or fourth-order accuracy in both space and time, and let the extension operator be used with $p=2$ or $p=4$. Consider the computed solution for the single- and composite-domain parabolic problems, with error measured in the maximum norm by the errors defined for the single or composite domain.

High-order accuracy conjecture. The error in the computed solution has second- and fourth-order accuracy in the maximum norm for the single- and composite-domain parabolic problems, respectively.

This expectation is motivated by the accuracy of the underlying discretization and extension operator, together with established error estimates and convergence results for the Difference Potentials Method for general linear elliptic boundary-value problems on smooth domains. The statement concerns the anticipated convergence of the method for implicitly defined geometries.