Approximation of partial differential equations on compact resistance spaces

TL;DR

This paper studies how solutions to linear PDEs on resistance spaces can be approximated by graph and metric graph models, establishing convergence results under various conditions.

Contribution

It provides new theoretical results on the uniform convergence of solutions to PDEs on resistance spaces via graph and metric graph approximations.

Findings

Solutions have accumulation points with respect to uniform convergence.

Convergence of solutions occurs when coefficients converge suitably.

Results apply to both discrete and metric graph approximations.

Abstract

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.