We show a norm convergence result for the Laplacian on a class of pcf self-similar fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional weighted graph Laplacians.As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.Our analysis works for a class of pcf self-similar sets (see Definition 3.1) which we call here fractals approximable by finite weighted graphs (see Definition 3.4). For such a fractal K there is a sequence (G m ) m∈N 0 of nested graphs G m = (V m , E m ) (i.e., V m ⊂ V m+1 ⊂ K) and conductances (i.e., edge weights) c e,m > 0 of the edges e ∈ E m , such that V * = m V m is dense in K, (see e.g. Figure 1 for the pentagasket with all five fixed points as boundary vertices) together with a compatible and self-similar sequence of graph energies (E m ) m (see Definitions 3.2 and 3.3) given byThe compatibility roughly means that E m (ϕ) agrees with the energy of E m+1 (h) where h : V m+1 −→ C is the harmonic extension of ϕ : V m −→ C. The harmonic extension h has the property that it minimises the energy E m (u) among all extensions u : V m+1 −→ C with u Vm = ϕ, see (3.2).

We define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.

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The purpose of this article is to prove that-under reasonable assumptions-the canonical energy form on a graph-like manifold is quasi-unitarily equivalent with the energy form on the underlying discrete graph. Then we will apply this to approximate the standard energy form on the Sierpiński gasket by a family of energy forms on suitable graph-like manifolds.

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