Approximation of partial differential equations on compact resistance spaces

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.


Introduction
For several classes of fractal spaces, such as for instance p.c.f. self-similar sets, [12, 60-63, 74, 75], classical Sierpinski carpets, [9,11], certain Julia sets, [89], Laaksø spaces, [90], diamond lattice fractals, [1,3,35], and certain random fractals, [33,34,79], the existence of resistance forms in the sense of [64,66] has been proved. This allows to establish a Dirichlet form based analysis, [15,27,78], with respect to a given volume measure, and in particular, studies of partial differential equations on fractals, [8,63,92]. These results and many later developments based on them are motivated by a considerable body of modern research in physics suggesting that in specific situations fractal models may be much more adequate than classical ones. The difficulty in this type of analysis comes from the fact that on fractals many tools from traditional calculus (and even many tools used in the modern theory of metric measure spaces, see e.g., [29,30]) are not available.
For fractal counterparts of equations of linear second order equations, [22,31], that do not involve first order terms -such as classical Poisson or heat equations for Laplacians on fractals -many results are known, [8,55,63,65,92], and there are also studies of related semilinear equations without first order terms, [24,25]. More recently fractal counterparts for equations involving first order terms have been suggested, [46,[49][50][51], and a few more specific results have been obtained, see for instance [47,77]. The discussion of first order terms is of rather abstract nature, because on most fractals there is no obvious candidate for a gradient operator; instead, it has to be constructed from a given bilinear form in a subsequent step, [17,18,49,54]. (An intuitive argument why this construction cannot be trivial is the fact that for self-similar fractals, endowed with natural Hausdorff type volume measures, volume and energy are typically singular, [14,[40][41][42].) For a study of, say, counterparts of second order equations, [31,Section 8], involving abstract gradient and divergence terms, it therefore seems desirable to establish auxiliary results which indicate that the equations have the correct physical meaning.
In this article we consider analogs of linear elliptic and parabolic equations with first order terms on locally compact separable resistance spaces, [64,66]. We wish to point out that we use the word 'elliptic' in a very broad (quadratic form) sense -the principal parts of our operators should rather be seen as fractal generalizations of hypoelliptic operators. Under suitable assumptions the equations admit unique weak respectively semigroup solutions, Corollaries 3.2 and 3. 3. We prove that if the resistance space is compact and we are given bounded sequences of coefficients, the corresponding solutions have uniform accumulation points, Corollary 4.3. If the sequences of coefficients converge, then the corresponding solutions converge in the L 2sense and uniformly along subsequences, Theorem 4.1. For local resistance forms on finitely ramified sets, [54,96], we introduce an approximation scheme along varying spaces general enough to accommodate both discrete and metric graph approximations. If the coefficients are bounded in a suitable manner, extensions of linearizations of solutions to the equations on the approximating spaces have uniform accumulation points on the target space, Corollary 5.5. If the coefficients are carefully chosen, the solutions converge in a suitable L 2 -sense and the mentioned extensions converge uniformly along subsequences, Theorem 5.1. Combining these results, we obtain an approximation result for quite general coefficients, Theorem 5.2.
For resistance forms on discrete and metric graphs the abstract gradient operators admit more familiar expressions, Examples 2.1 and 2.2, and the bilinear forms associated with linear equations can be understood in terms of the well-known analysis on graphs and metric graphs, Examples 3.1 and 3.2, see for instance [32,59] and [81,85]. The approximation scheme itself is of first order in the sense that it relies on the use of piecewise linear respectively piecewise harmonic functions, and it resembles familiar finite element methods. One motivation to use this approach is that pointwise restrictions of piecewise harmonic functions on, say, the Sierpinski gasket, are of finite energy on approximating metric graphs, [47], but for general energy finite functions on the Sierpinski gasket this is not true -the corresponding trace spaces on the metric graphs are fractional Sobolev spaces of order less than one, see for instance [93] and the reference cited there. Of course first order approximations have a certain scope and certain limitations. But keeping these in mind, we can certainly view our results as a strong first indication that the abstractly formulated equations on the target space have the desired physical meaning, because their solutions appear as natural limits of solutions to similar equations on more familiar geometries, where they are better understood. The established approximation scheme also provides a computational tool which could be used for numerical simulations. Our results hold under minimal assumptions on the volume measure on the target space. For instance, in the situation of p.c.f. self-similar structures it is not necessary to specialize to self-similar Hausdorff measures, [8,63], or to energy dominant Kusuoka type measures, [41,42,74,75].
In [94,Section 6] a finite element method for a Poisson type equation on p.c.f. self-similar fractals was discussed, and the use of an equivalent scalar product and a related orthogonal projection made it possible to regard the approximation itself as the solution of a closely related equation. For equations involving divergence and gradient terms one cannot hope for a similarly simple mechanism. On the other hand, the construction of resistance forms itself is based on discrete approximations, [60][61][62][63], and in symmetric respectively self-adjoint situations this can be used to obtain approximation results on the level of resistance forms, [19], or Dirichlet forms, [86,87]. In the latter case the dynamics of a partial differential equation of elliptic or parabolic type for self-adjoint operators comes into play, and it can be captured using spectral convergence results, [80,88], possibly along varying Hilbert spaces, [76,85]. The equations we have in mind are governed by operators that are not necessarily symmetric, but under some conditions on the coefficients they are still sectorial, [58,78]. This leads to the question of how to implement similar types on convergences for sectorial operators, and one arrives to a situation similar to those in [82] or [95]. The main difficulty is how to correctly implement the convergence of drift and divergence terms. With [47,86,87] in mind a possible first reflex is to try to verify a type of generalized norm resolvent convergence as in [82], and to do so the first order terms would have to fit the estimate in their Definition 2.3, where particular (2.7e) is critical. For convergent series of drift and divergence coefficients, [49], on a single resistance space one can establish this estimate with trivial identification operators (as adressed in their Example 2.5), but in the case of varying spaces the interaction of identification operators with the first order calculus seems too difficult to handle. The convergence results in [95,Section 4] use the variational convergence studied in [97,98], which generalizes the Mosco type convergence, [80], for generalized forms, [38], to the setup of varying Hilbert spaces, [76], and encodes a generalization of strong resolvent convergence. Also in the present article this variational convergence is used as a key tool: We verify the adequate Mosco type convergence along varying spaces of the bilinear forms associated with the equations, and by [38] and [97,98] we can then conclude the L 2 -type convergence of the solutions, see Theorem A.1. A significant difference between [95,Section 4] and our results is the way the first order terms are handled. There the approach from [5] is used, which relies heavily on having a carré du champ operator, [15]. But this is an assumption which we wish to avoid, because -as mentioned above -interesting standard examples do not satisfy it. The target spaces for the approximation result along varying spaces that we implement are assumed to be finitely ramified sets, [54,96], endowed with local regular resistance forms, [64,66]. This class of fractals contains many interesting examples, [1,3,33,34,36,79,89,90], and in particular, p.c.f. self-similar fractals, [63], but it does not contain Sierpinski carpets, [9,11]. The cell structure of a finitely ramified set allows a transparent use of identification operators based on piecewise harmonic functions. From a technical point of view the key property of resistance spaces that energy finite functions are continuous compensates the possible energy singularity of a given volume measure to a certain extent, in particular, we can use an inequality originally shown in [48] when handling the first order terms in the presence of an energy singular measure. Uniform energy bounds and the compactness of the space (in resistance metric) then allow to use Arzela-Ascoli type arguments to obtain subsequential limits in the sense of uniform convergence. Together with the L 2 -type limit statements produced by the variational convergence these limit points are then identified to be the solutions on the target space.
The use of variational convergence schemes in order to gain insight into dynamical phenomena on certain geometries is a robust and prominent idea, see for instance [56,57,76]. It was already a guiding theme in [80], and related results in different setups have been studied in a number of recent articles, see for instance [4,23,43,53,67,68,82,[84][85][86][87]95], just to mention a few. For fractal spaces variational schemes can provide a certain counterpart to homogenization: In the latter the effect of a complicated microstructure can be encoded in an equation for an effective material if the problem is viewed at a certain mesoscopic scale. In analysis on fractals it may not be possible to find such a scale, and it is desirable to have a more direct understanding of how the microstructure determines analysis. This typically leads to non-classical rescalings when passing from discrete to continuous or from smooth to fractal. Although the present study is written specifically for resistance spaces, some aspects of the approximation scheme in Section 5 might also provide a rough guideline for the implementation of schemes along varying spaces for non-symmetric local or non-local operators on non-resistance spaces, and there are plenty of problems for which no results of this type are available.
We proceed as follows. In Section 2 we recall basics from the theory of resistance forms and explain items of the related first order calculus. We discuss bilinear forms including drift and divergence terms in Section 3, and follow standard methods, [22,26,31], to state existence, uniqueness and energy estimates for weak solutions to elliptic equations and (semigroup) solutions to parabolic equations. In Section 4 we prove convergence results for equations on a single compact resistance space. We first discuss suitable conditions on the coefficients, then accumulation points and then strong resolvent convergence. Section 5 contains the approximation scheme along varying spaces for finitely ramified sets. We first state the basic assumptions and record some immediate consequences, then survey conditions on the coefficients and finally state the accumulation and convergence results. Section 6 discusses discrete approximations (Subsection 6.1), including classes of examples, metric graph approximations (Subsection 6.2), and short remarks on possible generalizations. Section 7 contains an auxiliary result on the restriction of vector fields for finitely ramified sets.
We follow the habit to write E(u) for E(u, u) if E is a bilinear quantity depending on two arguments and both arguments are the same.

Resistance forms and first order calculus
We recall the definition of resistance form, due to Kigami, see [63,Definition 2.3.1] or [64,Definition 2.8]. By ℓ(X) we denote the space of real valued functions on a set X.
is a linear subspace of ℓ(X) containing the constants and E is a non-negative definite symmetric bilinear form on F with E(u) = 0 if and only if u is constant.
To R one refers as the resistance metric associated with (E, F ), [64,Definition 2.11], and to the pair (X, R), which forms a metric space, [64, Proposition 2.10], we refer as resistance space. All functions u ∈ F are continuous on X with respect to the resistance metric, more precisely, we have For any finite subset V ⊂ X the restriction of (E, F ) to V is the resistance form (E V , ℓ(V )) defined by where a unique infimum is achieved. If V 1 ⊂ V 2 and both are finite, then ( We assume X is a nonempty set and (E, F ) is a resistance form on X so that (X, R) is separable. Then there exists a sequence (V m ) m of finite subsets V m ⊂ X with V m ⊂ V m+1 , m ≥ 1, and m≥0 V m dense in (X, R). For any such sequence (V m ) m we have as proved in [64, Proposition 2.10 and Theorem 2.14]. Note that for any u ∈ F the sequence (E Vm (u)) m is non-decreasing. Each E Vm is of the form with constants c(m; p, q) ≥ 0 symmetric in p and q. We further assume that (X, R) is locally compact and that (E, F ) is regular, i.e., the space F ∩ C c (X) is uniformly dense in the space C c (X) of continuous compactly supported functions on (X, R), see [66, Definition 6.2]. Definition 2.1 (v) implies that F ∩ C c (X) is an algebra under pointwise multiplication and see [66,Lemma 6.5].
To introduce the first order calculus associated with (E, F ), let ℓ a (X × X) denote the space of all real valued antisymmetric functions on X × X and write (6) (g · v)(x, y) := g(x, y)v(x, y), x, y ∈ X, for any v ∈ ℓ a (X × X) and g ∈ C c (X), where g(x, y) := 1 2 (g(x) + g(y)).
Obviously g · v ∈ ℓ a (X × X), and (6) defines an action of C c (X) on ℓ a (X × X), turning it into a module. By d u : F ∩ C c (X) → ℓ a (X × X) we denote the universal derivation, and by deviating slightly from the notation used in [47], the submodule of ℓ a (X × X) of finite linear combinations of functions of form g·d u f . A quick calculation shows that for f, g ∈ F ∩C c (X) we have d u (f g) = f ·d u g+g·d u f .
On Ω 1 a (X) we can introduce a symmetric nonnegative definite bilinear form ·, · H by extending linearly in both arguments, respectively, and we write · H = ·, · H for the associated Hilbert seminorm. In Proposition 2.1 below we will verify that the definition of ·, · H does not depend on the choice of the sequence (V m ) m .
We factor Ω 1 a (X) by the elements of zero seminorm and obtain the space Ω 1 a (X)/ ker · H . Given an element Completing Ω 1 a (X)/ ker · H with respect to · H we obtain a Hilbert space H, we refer to it as the space of generalized L 2 -vector fields associated with (E, F ). This is a version of a construction introduced in [17,18] and studied in [13, 48-52, 54, 77], see also the related sources [20,29,30,99]. The basic idea is much older, see for instance [15,Exercise 5.9], it dates back to ideas of Mokobodzki and LeJan.
The action (6) induces an action of C c (X) on H: (6) and (9) imply it follows that the definition of g · v is correct. Given f ∈ F ∩ C c (X), we denote the H-equivalence class of the universal derivation d u f as in (7) by ∂f . By the preceding discussion we observe [g · d u f ] H = g · ∂f for all f ∈ F ∩ C c (X) and g ∈ C c (X). It also follows that the map f → ∂f defines a derivation operator which satisfies the identity ∂f 2 H = E(f ) for any f ∈ F ∩ C c (X) and the Leibniz rule ∂(f g) = f · ∂g + g · ∂f for any f, g ∈ F ∩ C c (X).
To show the independence of ·, · H of the choice of the sequence (V m ) m in (9) and to formulate later statements, we consider energy measures. For f ∈ F ∩ C c (X) there is a unique finite Radon measure ν f on X satisfying the energy measure of f , see for instance [55,65,75,96] and or [27,[39][40][41][42]45]. It is not difficult to see that for any f ∈ F ∩ C c (X) and g ∈ C c (X) we have Mutual energy measures ν f1,f2 for f 1 , f 2 ∈ F ∩ C c (X) are defined using (11) and polarization. According to the Beurling-Deny decomposition of (E, F ), see [2, Théorème 1] (or [27, Section 3.2] for a different context), there exist a uniquely determined symmetric bilinear form E c on F ∩ C c (X) satisfying E c (f, g) = 0 whenever f ∈ F ∩ C c (X) is constant on an open neighborhood of the support of g ∈ F ∩ C c (X) and a uniquely determined symmetric nonnegative Radon measure J on X × X \ diag such The form E c is called the local part of E, and by ν c f we denote the local part of the energy measure of a function f ∈ F ∩ C c (X), i.e. the finite Radon measure (uniquely) defined as in (11) with E c in place of E. From (11) and (13) it is immediate that Proposition 2.1. Suppose that closed balls in (X, R) are compact. Then for any f 1 , f 2 ∈ F ∩ C c (X) and g 1 , g 2 ∈ C c (X) we have In particular, the definition of the bilinear form ·, · H is independent of the choice of the sets V m .
Proof. Standard arguments show that for all v ∈ C c (X × X \ diag) we have see for instance [27,Section 3.2]. The particular case v = d u f , together with (13), then implies that for any f ∈ F ∩ C c (X). We claim that given such f and g ∈ C c (X), This follows from (11) and (16) and the fact that which can be seen following the arguments in the proof of [47, Lemma 3.1]. Combining (15), applied to v = g · d u f , and (17), we obtain the desired result by polarization.
As a consequence of Proposition 2.1 and dominated convergence we can define g · v for all v ∈ H and g ∈ C b (X) and (10) remains true for such v and g. Note also that if v 1 , v 2 ∈ H and g ∈ C b (X) then In the special cases of finite graphs, [32,59], and compact metric graphs, [28, 69-72, 81, 85], the space H and the operator ∂ appear in a more familiar form.
Examples 2.1. If (V, ω) is a finite simple weighted (unoriented) graph, [32], then is a resistance form on the finite set V , and it makes it a compact resistance space. In this case H is isometrically isomorphic to the space ℓ 2 a (V ×V \diag, ω) of real-valued antisymmetric functions on V ×V \diag, endowed with the usual ℓ 2 -scalar product, and for any f ∈ ℓ(V ) the gradient ∂f ∈ H of f is the image of d u f ∈ ℓ 2 a (V × V \ diag, ω) under this isometric isomorphism, see for instance [44,Section 3]. Examples 2.2. Let (V, E) be a finite simple (unoriented) graph and (l e ) e∈E a finite sequence of positive numbers. Consider the metric graph Γ obtained by identifying each edge e ∈ E with an oriented copy of the interval (0, l e ) and considering different copies to be joined at the vertices the respective edges have in common. Then the set X Γ = V ∪ e∈E e, endowed with a natural topology, becomes a compact metric space. For each u ∈ C(X Γ ) let and u e is the restriction of u to e ∈ E. IfẆ 1,2 (X Γ ) denotes the space of all u ∈ C(X Γ ) such that E(u) < +∞ then (E Γ ,Ẇ 1,2 (X Γ )) is a resistance form making X Γ a compact resistance space. The space H is isometrically isomorphic to e∈E L 2 (0, l e ), and for any f ∈Ẇ 1,2 (X Γ ) the gradient ∂f ∈ H is the image under this isometric isomorphism of (f ′ e ) e∈E , where f ′ e ∈ L 2 (0, l e ) denotes the usual first derivative of f e , seen as a function on (0, l e ). For more precise descriptions and further details see [13,54]. In Subsection 6.2 we consider a scaled variant of this construction as in [47].
Remark 2.1. For convenience the above construction of the space H and the operator ∂ is formulated for resistance spaces. However, we wish to point out that the original construction does not need the specific properties of a resistance space, it can be formulated for Dirichlet forms in very high generality, [17].

Linear equations of elliptic and parabolic type
The considerations in this section are straightforward from standard theory for partial differential equations, [31,Chapter 8], and Dirichlet forms, [27], see for instance [26].
Let (E, F ) be a resistance form on a nonempty set X so that (X, R) is separable and locally compact and assume that (E, F ) is regular. Let µ be a Borel measure on (X, R) such that for any x ∈ X and R > 0 we have 0 < µ(B(x, R)) < +∞. Then by [66,Theorem 9.4] the form (E, F ∩ C c (X)) is closable on L 2 (X, µ) and its closure, which we denote by (E, D(E)), is a regular Dirichlet form. In general we have D(E) ⊂ F ∩ L 2 (X, µ), and in the special case that (X, R) is compact, D(E) = F , [66,Section 9]. Given α > 0 we write and we use an analogous notation for other bilinear forms. Recall that we also write E(f ) to denote E(f, f ) and similarly for other bilinear quantities. By the closedness of (E, D(E)) the derivation ∂, defined as in the preceding section, extends to a closed unbounded linear operator ∂ : L 2 (X, µ) → H with domain D(E), we write Im ∂ for the image of D(E) under ∂. The adjoint operator (∂ * , D(∂ * )) of (∂, D(E)) can be interpreted as minus the divergence operator, and for the generator (L, D(L)) of (E, D(E)) we have ∂f ∈ D(∂ * ) whenever f ∈ D(L), and in this case, Lf = −∂ * ∂f .
3.1. Closed forms. We call a symmetric bounded linear operator a : H → H a uniformly elliptic (in the sense of quadratic forms) if there are universal constants 0 < λ < Λ such that (19) λ As mentioned in the introduction, the phrase 'uniformly elliptic' is interpreted in a wide sense, and (19) rather corresponds to a sort of energy equivalence, see for instance [10,Definition 2.17]. We follow [26] and say that an element b ∈ H is in the Hardy class if there are constants δ(b) ∈ (0, ∞) and γ(b) ∈ [0, ∞) such that (20) g . Given uniformly elliptic a as in (19), b,b ∈ H in the Hardy class and c ∈ L ∞ (X, µ) we consider the bilinear form on F ∩ C c (X) defined by We say that a densely defined bilinear form (Q, D(Q)) on L 2 (X, µ) is semibounded if there exists some . If in addition (D(Q), Q C+1 ) is a Hilbert space, where Q denotes the symmetric part of Q, defined by then we call (Q, D(Q)) a closed form. In other words, we call (Q, D(Q)) a closed form if ( Q, D(Q)) is a closed quadratic form in the sense of [88, Section VIII.6]. We say that a closed form (Q, D(Q)) is sectorial if there is a constant K > 0 such that where C is as above. In other words, we call a closed form (Q, D(Q)) sectorial if (Q C , D(Q)) is a coercive closed form in the sense of [78,Definition 2.4].
The following proposition follows from standard estimates and (20), we omit its proof.
(i) Assume that a : H → H is symmetric and satisfies (19), c ∈ L ∞ (X, µ) and b,b ∈ H are in the Hardy class and such that Then (Q, F ∩ C c (X)) is closable on L 2 (X, µ), and its closure (Q, D(E)) is a sectorial closed form. (ii) If in addition c is such that (24) c 0 := ess inf Remark 3.1. These conditions are chosen for convenience, we do not claim their optimality. Standard estimates using integrability conditions for vector fields, as for instance used in [95], do not apply unless one assumes that energy measures are absolutely continuous with respect to µ, an assumption we wish to avoid.
Suppose that the hypotheses of Proposition 3.1 (i) are satisfied. Let (L (Q) , D(L (Q) )) denote the infinitesimal generator of (Q, D(E)), that is, the unique closed operator on L 2 (X, µ) associated with (Q, D(E)) by the identity A direct calculation shows the following.
Corollary 3.1. Let the hypotheses of Proposition 3.1 (i) and (ii) be satisfied, let notation be as there and set Then the generator (L (Q) , D(L (Q) )) satisfies the sector condition

3.2.
Linear elliptic and parabolic problems. Suppose throughout this subsection that a, b,b and c satisfy the hypotheses of Proposition 3.1 (i) and (ii). It is straightforward to formulate equations of elliptic type. Given f ∈ L 2 (X, µ), we say that u ∈ L 2 (X, µ) is a weak solution to (29) L Q u = f if u ∈ D(E) and Q(u, g) = − f, g L 2 (X,µ) for all g ∈ D(E).
Remark 3.2. Formally, the generator (L Q , D(L Q )) of (Q, D(E)) has the structure so that equation (29) is seen to be an abstract version of the elliptic equation It follows from the lower estimate in (25) that the Green operator G Q = (−L Q ) −1 of L Q exists as a bounded linear operator G Q : L 2 (X, µ) → L 2 (X, µ) and satisfies (30) Q is the unique weak solution to (29). It satisfies Remark 3.3. The constant in (31) is chosen just for convenience. The only fact that matters is that it may be chosen in a way that depends monotonically on c 0 .
Proof. The first part is clear, the second follows from (30), Cauchy-Schwarz and because for any 0 < ε ≤ c 0 /2 with c 0 as in (24) the operator L Q + ε generates a strongly continuous contraction semigroup, so that Remark 3.4. If c ∈ L ∞ (X, µ) does not satisfy (24), one can at least solve equations where c 1 > 0 is such that with c 0 defined as in (24) one has c 0 + c 1 > 0. The sectorial closed form satisfies (25), (26), (28) and (27) with c 0 + c 1 and c L ∞ (X,µ) + c 1 in place of c 0 and c L ∞ (X,µ) .
Related parabolic problems can be discussed in a similar manner. Givenů ∈ L 2 (X, µ) we say that a function u : (0, +∞) → L 2 (X, µ) is a solution to the Cauchy problem Remark 3.5. Problem (34) is an abstract version of the parabolic problem Let (T Q t ) t>0 denote the strongly continuous contraction semigroup on L 2 (X, µ) generated by the operator L Q . The following is standard.
where C K > 0 is a constant depending only on the sector constant K in (28).
Proof. Again the first part of the Corollary is standard. To see (35) recall that the operator (L Q , D(L Q )) satisfies the sector condition (28). Consequently the semigroup (T Q t ) t>0 generated by (L Q + ε, D(L Q )) extends to a holomorphic contraction semigroup on the sector {z ∈ C : | Im z| ≤ K −1 Re z}, see for instance [ (25) zero is contained in the resolvent set of L Q . This implies that for any t > 0 we have for some C K ∈ (0, ∞) depending only on the sector constant K, as an inspection of the classical proofs of (36) shows, see for instance [  It is a trivial observation that if a ∈ C(X) satisfies then a, interpreted as a bounded linear map v → a · v from H into itself, satisfies (19). Our main interest is to understand the drift terms and therefore we restrict attention to coefficients a of form (37) in the following sections. Note that under condition (37) the function a may also be seen as a conformal factor, [7].
Remark 3.7. A discussion of more general diffusion coefficients a should involve suitable coordinates, see [41,52,96]. In view of the fact that natural local energy forms on p.c.f. self-similar sets have pointwise index one, [13,42,75], assumption (37) does not seem to be unreasonably restrictive for this class of fractal spaces.
Examples 3.1. In the setup of Examples 2.1 and with a given volume function µ : V → (0, +∞) we obtain, accepting a slight abuse of notation, for all f, g ∈ ℓ(V ) and any given coefficients a, c ∈ ℓ(V ) and Examples 3.2. Suppose we are in the same situation as in Examples 2.1 and µ is a finite Borel measure on X Γ that has full support and is equivalent to the Lebesgue-measure on each individual edge. Then, abusing notation slightly, where u e denoted the restricition to e ∈ E in the a.e. sense of an integrable function on X Γ .

Convergence of solutions on a single space
In this section we define bilinear forms Q (m) on L 2 (X, µ) by replacing a, b andb in (21) by coefficients a m b m andb m that may vary with m. To keep the exposition more transparent and since it is rather trivial to vary it, we keep c fixed. We consider the unique weak solutions to elliptic problems (29) and unique solutions at fixed positive times of parabolic problems (34) with these coefficients. For a sequence (a m ) m satisfying (19) uniformly in m, bounded sequences (b m ) m and (b m ) m and small enough c, we can find accumulation points with respect to the uniform convergence on X of these solutions, and these accumulation points are elements of F , Corollary 4.3. If coefficients a, b,b and c are given and the sequences (a m ) m , (b m ) m and (b m ) m converge to a, b andb, respectively, then we can conclude the uniform convergence of the solutions to the respective solutions of the target problem, Theorem 4.1.

4.1.
Boundedness and convergence of vector fields. As in the preceding section we assume that (X, R) is separable and locally compact, that (E, F ) is regular and that µ is a Borel measure on (X, R) such that for any x ∈ X and R > 0 we have 0 < µ(B(x, R)) < +∞.
Under a mild geometric assumption on µ any vector field b ∈ H satisfies the Hardy condition. We say that µ has a uniform lower bound V if V is an non-decreasing function V : (0, +∞) → (0, +∞) so that The following proposition is a partial refinement of [48,Lemma 4.2].
Proposition 4.1. Suppose that µ has the uniform lower bound V . Then for any g ∈ F ∩ C c (X), any b ∈ H and any M > 0 we have where V is the non-decreasing function In particular, any b ∈ H is in the Hardy class, and for any M > 0 it satisfies the estimate (20) A proof of an inequality of type (39) had already been given in [48,Lemma 4.2], but the function V had not been specified and an unnecessary metric doubling assumption had been made. We comment on the necessary modifications.
As in [48] we can use (1) to see that for any j and any x ∈ B j we have and using (10) we arrive at the claimed inequality.
We record two consequences of Proposition 4.1. The first states that if the norms of vector fields in a sequence are uniformly bounded then we may choose uniform constants in the Hardy condition (20).
The second consequence is a continuity statement.

Corollary 4.2.
Suppose that µ has a uniform lower bound.
Proof. This is immediate from the definition of the function V in Proposition 4.1 and the fact that the uniform lower bound V of µ is strictly positive and increasing.

Accumulation points.
For the rest of this section we assume that (E, F ) is a regular resistance form on a nonempty set X so that (X, R) is compact, and that µ is a finite Borel measure on (X, R) with a uniform lower bound V . For each m let a m ∈ C(X) satisfy (37) with the same constants 0 < λ < Λ. Suppose M > 0 is large enough so that Let γ M be as in (40), letγ M defined in the same way with theb m replacing the b m and suppose that c ∈ L ∞ (X, µ) is such that Then by Proposition 3.1 and Corollary 4.1 the forms are sectorial closed forms on L 2 (X, µ). They satisfy (25) (26). Their generators (L Q (m) , D(L Q (m) )) satisfy the sector conditions (28) with the same sector constant K. As a consequence we observe uniform energy bounds for the solutions of (29) and (34). We write Q (m),α for the form defined like E α in (18) but with Q (m) in place of E.
Proposition 4.2. Let a m , b m ,b m and c be as above such that (41) and (42) hold.
(i) If f ∈ L 2 (X, µ) and u m is the unique weak solution to (29) with L Q (m) in place of L, then we have Proof. Since (42) and (28) hold with the same constants c 0 and K for all m, the statements follow from Corollaries 3.2 and 3.3.
The compactness of X implies the existence of accumulation points in C(X).  (41) and (42) are satisfied.
(i) If f ∈ L 2 (X, µ) and u m is the unique weak solution to (29) with L Q (m) in place of L Q , then each subsequence of (u m ) m has a subsequence converging to a limit u ∈ F uniformly on X. (ii) Ifů ∈ L 2 (X, µ) and u m is the unique solution to (34) with L Q (m) in place of L Q , then for each t > 0 each subsequence of (u m (t)) m has a further subsequence converging to a limit u t ∈ F uniformly on X.
At this point we can of course not claim that the C(X)-valued function t → u t has any good properties or significance.
Proof. Since all Q (m) satisfy (25) with the same constants, Proposition 4.2 implies that sup m E 1 (u m ) < +∞. By [66, Lemma 9.7] the embedding F ⊂ C(X) is compact, hence (u m ) m has a subsequence that converges uniformly on X to a limit u. To see that u ∈ F , note that also this subsequence is bounded in F and therefore has a further subsequence that converges to a limit w ∈ F weakly in L 2 (X, µ), as follows from a Banach-Saks type argument. This forces w = u. Statement (ii) is proved in the same manner. 4.3. Strong resolvent convergence. Let (E, F ) and µ be as in the preceding subsection. Let a ∈ F be such that (37) holds with constants 0 < λ < Λ and let (a m ) m ⊂ C(X) be such that Without loss of generality we may then assume that also the functions a m satisfy (37) with the very same constants 0 < λ < Λ. Suppose M > 0 is large enough so that Note that this implies (41). Let γ M be as in (40) andγ M similarly but with theb m , and suppose that c ∈ L ∞ (X, µ) satisfies (42). Let Q be as in (21) and Q (m) as in (43). The next theorem states that the solutions to (29) and (34) (44) and (45) hold. Then lim m L Q (m) = L Q in the strong resolvent sense, and the following hold.
(i) If f ∈ L 2 (X, µ), u and u m are the unique weak solutions to (29) and to (29) with L Q (m) in place of L Q , respectively, then lim m u m = u in L 2 (X, µ). Moreover, there is a sequence (m k ) k with m k ↑ +∞ such that lim k u m k = u uniformly on X. (ii) Ifů ∈ L 2 (X, µ), and u and u m are the unique solutions to (34) and to (34) with L Q (m) in place of L, then for any t > 0 we have lim m u m (t) = u(t) in L 2 (X, µ). Moreover, for any t > 0 there is a sequence (m k ) k with m k ↑ +∞ such that lim k u m k (t) = u(t) uniformly on X.
Proof. By [38, Theorem 3.1], the claimed strong resolvent convergence and the stated convergences in L 2 (X, µ) follow once we have verified the conditions in Definition A.2, see Theorem A.1 and Remark A.3 in Appendix A. The statements on uniform convergence then also follow using Corollary 4.3.
Without loss of generality we may assume that the function c ∈ L ∞ (X, µ) satisfies condition (42). If not, proceed similarly as in Remark 3.4 and replace c by c − c 1 , where c 1 > 0 is large enough so that with c 0 as defined as in (42) we have c 1 + c 0 > 0, and consider the forms (Q (m),c1 , F ) with generators (L Q (m) − c 1 , D(L Q (m) )). If lim m→∞ L Q (m) − c 1 = L Q − c 1 in the KS-generalized strong resolvent sense then also lim m→∞ L Q (m) = L Q in the KS-generalized strong resolvent sense. The statements on uniform convergence then follow using Corollary 4.3 and Remark 3.4, note that for all m and u ∈ F we have Thanks to (23), (24), (25) and (26) together with Proposition 4.1 and Corollaries 4.1 and 4.2 we can find a constant C > 0 such that for every sufficiently large m we have To verify condition (ii) in Definition A.2 suppose that (m k ) k be a sequence of natural numbers with what implies that (u k ) k has a subsequence (u kj ) j converging to u ∈ F weakly in F and uniformly on X, and such that its averages N −1 N j=1 u kj converge to u in F . Here the statement on uniform convergence is again a consequence of the compact embedding F ⊂ C(X), [66,Lemma 9.7]. Combined with the weak convergence in L 2 (X, µ) it follows that (u kj ) j converges weakly to u in (F / ∼, E). Moreover, using (13), the convergence of averages and the linearity of d u we may assume that (d u u kj ) j converges to d u u weakly in L 2 (X × X \ diag, J). As a consequence, we also have for all v ∈ F . Now let w ∈ F . Then we have Since c is kept fixed, the first summand on the right hand side of the inequality (47) is bounded by where we have used Cauchy-Schwarz and (10). By the hypotheses on the coefficients and the boundedness of (u kj ) j in energy and in uniform norm this converges to zero. The second summand on the right hand side of (47) is bounded by The last summand in this line obviously converges to zero, and also the second does, note that | (u kj − u) · b, ∂w H | ≤ u kj − u sup b H E(w) 1/2 by Cauchy-Schwarz and (10). By Proposition 2.1 we have Since ad u w L 2 (X×X\diag,J) ≤ a sup E(w) 1/2 , the double integral converges to zero by the weak convergence of (d u u kj ) j to d u u in L 2 (X × X \ diag, J). By (5) we have sup j E 1 (au kj ) 1/2 < +∞ and E 1 (wu kj ) 1/2 < +∞. Thinning out the sequence (u kj ) j once more we may, using the arguments above, assume that converges to zero. Together this implies that lim j ∂w, a · ∂(u kj − u) H = 0. Finally, note that by the Leibniz rule for ∂, As before we see easily that the second summand on the right hand side goes to zero. For the first, let b = ∂f + η be the unique decomposition ofb ∈ H into a gradient ∂f of a function f ∈ F and a 'divergence free' vector field η ∈ ker ∂ * . Then which converges to zero by the preceding arguments. Combining, we see that and since w ∈ F was arbitrary, this implies condition (ii) in Definition A.2.

Convergence of solutions on varying spaces
In this section we basically repeat the approximation program from Section 4, but now on varying resistance spaces. More precisely, we study the convergence of suitable linearizations of solutions to (29) and (34) on approximating spaces X (m) to solutions to these equations on X. We establish these results for the case that X is a finitely ramified set, [54,96], endowed with a local resistance form. Possible generalizations are commented on in Section 6.3.

Setup and basic assumptions.
We describe the setup we consider and the assumptions under which the results of this section are formulated. They are standing assumptions for all results in this section and will not be repeated in the particular statements.
We recall the notion of finitely ramified cell structures as introduced in [96, Definition 2.1].
Definition 5.1. A finitely ramified set X is a compact metric space which has a cell structure {X α } α∈A and a boundary (vertex) structure {V α } α∈A such that the following hold: (i) A is a countable index set; (ii) each X α is a distinct compact connected subset of X; Under these conditions the triple (X, {X α } α∈A , {V α } α∈A ) is called a finitely ramified cell structure.
We write V n = α∈An V α and V * = n≥0 V n , note that V n ⊂ V n+1 for all n.   [96,Theorem 2], that the restriction to V * of a function on X, continuous on w.r.t. the original topology, is continuous on (V * , R), that any R-Cauchy sequence converges in X w.r.t. the original topology, and that there is a continuous injective map θ : Ω → X which is the identity on V * . This allows to identify Ω with the R-closure X R of V * in X. The space (X R , R) is compact and locally connected. Let diam R (A) denote the diameter of a set A in (X R , R). From [96, Theorem 3 and its proof] it follows that (E, F ) is a regular resistance form on X R and that it is local: If f ∈ F is constant on an R-open neighborhood of the support (w.r.t. R) of g ∈ F , then E(f, g) = 0. We write H m (X) to denote the space of m-harmonic functions on X and write H m u := h m (u| Vm ), u ∈ F , for the projection from F onto H m (X). It is well known and can be seen as in [92,Theorem 1.4.4] that (48) lim and using (1) it follows that also lim m u − H m u sup = 0, where · sup denotes the supremum norm. Consequently the space is dense in F w.r.t. the seminorm E 1/2 and w.r.t. the supremum norm, and as discussed above, Assumption 5.1 ensures that H * (X) is also dense in C(X R ) (and also in the space of restrictions to X R of functions on X continuous in the original topology). We write H m (X)/ ∼ for the space of m-harmonic functions on X modulo constants. For each m the space H m (X)/ ∼ is a finite dimensional, hence closed subspace of (F / ∼, E), and since H m 1 = 1, the operator H m is easily seen to induce an orthogonal projection in (F / ∼, E) onto H m (X)/ ∼, which we denote by the same symbol. Clearly H * (X)/ ∼ is dense in (F / ∼, E). We now state the main assumptions under which we implement the approximation scheme. They are formulated in a way that simultaneously covers approximations schemes by discrete graphs and by metric graphs as discussed in Sections 6.1 and 6.2, respectively.
The following assumption requires E to be compatible with the cell structure in the following 'uniform' metric sense.
We now assume that (X (m) ) m is a sequence of subsets X (m) ⊂ X such that for each m ≥ 0 we have X (m) ⊂ X (m+1) and X (m) = α∈Am X (m) α where for any α ∈ A m the set X For any m ≥ 0 let now (E (m) , F (m) ) be a resistance form on X (m) so that (X (m) , R (m) ) is topologically embedded in (X R , R). We also assume that the spaces (X (m) , R (m) ) are compact, this implies that the resistance forms (E (m) , F (m) ) are regular. By ν (m) f we denote the energy measure of a function f ∈ F (m) , defined as in (11) with (E (m) , F (m) ) in place of (E, F ). The energy measures ν (m) f may be interpreted as Borel measures on X.
Remark 5.1. For spaces, forms, operators, coefficients and measures indexed by m and connected to X and the form (E, F ) we will use a subscript index m, similar objects corresponding to the spaces X (m) and the forms (E (m) , F (m) ) will be indexed by a superscript (m), unless stated otherwise. For functions we will generally use a subscript index.
We make some further assumptions. The first expresses a connection between the resistance forms in terms of m-harmonic functions. Hm(u)| X (m) , u ∈ F , in the sense of weak convergence of measures on X.
As a trivial consequence of (50) we have Remark 5.2. For approximations by discrete graphs (50) follows from (51) and (12). For metric graph approximations (50) is verified in Subsection 6.2 below, the use of products in (11) hinders a direct conclusion of (50) from (51).  ) are regular Dirichlet forms on L 2 (X R , µ) and L 2 (X (m) , µ (m) ), and the Dirichlet form (E, F ) is strongly local in the sense of [27]. To ease notation we will frequently view µ and µ (m) as measures on X and for instance write L 2 (X, µ) instead of L 2 (X R , µ).
We make an assumption on the connection between the spaces L 2 (X, µ) and L 2 (X (m) , µ (m) ) and its consistency with the projections and pointwise restrictions. By Ext m : H m (X (m) ) → H m (X) we denote the inverse of the bijection u → u| X (m) from H m (X) onto H m (X (m) ).
(i) The measures µ and µ (m) admit a uniform lower bound in the following sense: There is a nonincreasing function V : N → (0, +∞) such that for any m we have µ(X α ) ≥ V (m), α ∈ A m , and moreover, µ (m) (X Let H and H (m) denote the spaces of generalized L 2 -vector fields associated with (E, F ) and (E (m) , F (m) ), respectively. The corresponding gradient operators we denote by ∂ and ∂ (m) . If a, b,b and c satisfy the hypotheses of Proposition 3.1 (i) then defines a sectorial closed form (Q, F ) on L 2 (X, µ). If a and c are suitable continuous functions on X R and b, b, b (m) andb (m) are vector fields of a suitable form, then we can define sectorial closed forms (Q (m) , F (m) ) on the spaces L 2 (X (m) , µ (m) ), respectively, by In Subsection 5.4 below we observe that under simple boundedness assumptions the solutions of (29) and (34) (for fixed t > 0) associated with the forms Q (m) on the spaces X (m) accumulate in a suitable sense, see Proposition 5.2. In Theorem 5.1 in Subsection 5.5 we then conclude that they actually converge to the solutions to the respective equation associated with the form Q, as announced in the introduction. In the preparatory Subsections 5.2 and 5.3 we record some consequences of the assumptions and discuss possible choices for b,b, b (m) andb (m) .

5.2.
Some consequences of the assumptions. We record some consequences of the above assumptions and begin with well known conclusions.
(i) For any p, q ∈ V m we have R (m) (p, q) = R(p, q). In particular, diam R (V α ) = diam R (m) (V α ) for any m ≥ n and α ∈ A n . (ii) We have diam R (X α ∩X R ) = diam R (V α ) for any n and α ∈ A n , and diam R (m) (X (n) Proof. To see (i) note that for any p, q ∈ V m we have, by a standard conclusion and using (49) and (52), If the first statement (ii) were not true we could find p ∈ X α ∩ V * and q ∈ (X α ∩ V * ) \ V α such that R(p, q) > R(p, q ′ ) for all q ′ ∈ V α . This would imply that there exists some u ∈ H n (X) with u(p) = 0 and E(u) = 1 such that u(q) 2 < u(q ′ ) 2 for all q ′ ∈ V α . However, this contradicts the maximum principle. The second statement follows similarly.
Also the following is due to Assumption 5.3.
Corollary 5.1. For any f 1 , f 2 ∈ H n (X) and g 1 , g 2 ∈ C(X R ) we have If all E (m) 's are local then by Proposition 2.1 we have for all f ∈ H n (X) and g ∈ C(X R ), where ν (m),c f denotes the local part of the energy measure of f with respect to (E (m) , F (m) ), and by (50) this converges to Suppose now that the E (m) 's have nontrivial jump measures J (m) . If f ∈ H n (X) and g ∈ H n ′ (X) have disjoint supports then by Proposition 2.1, the locality of E (m),c , (49) and the locality of E we have Given f, g ∈ C(X R ) with disjoint supports, we can, by the proof of [96, Theorem 3], find sequences of functions (f j ) j and (g j ) j from H * (X) approximating f and g uniformly and disjoint compact sets K(f ) ⊂ X R and K(g) ⊂ X R such that all f j and g j are supported in K(f ) and K(g), respectively. Therefore (61) and the arguments used in the proof of [27, Theorem 3.2.1] imply that lim m J (m) = 0 vaguely on X × X \ diag. For functions f ∈ H n (X) and g ∈ C(X) we therefore have as can be seen using the arguments in the proof of [47, Lemma 3.1]. On the other hand we have for such f and g by (50) and (14). Combining and taking into account Proposition 2.1 we can conclude that , from which the stated result follows by polarization.
Another consequence, in particular of Assumption 5.5, is the convergence of the L 2 -spaces and the energy domains in the sense of Definition A.1.

Corollary 5.2.
(i) We have in the KS-sense with identification operators Φ m as above. Proof. Statement (i) is immediate from (56). To see statement (ii) let u ∈ F . If x 0 ∈ V 0 is fixed, we have H m u(x 0 ) = u(x 0 ) for any m and therefore, by (1) and (48), Using (55), we obtain lim m Φ m H m u − Φ m u L 2 (X,µ) = 0, and combining with (58) and (56), Together with (51) this shows that lim m E In the sequel we will say 'KS-weakly' resp. 'KS-strongly' if we refer to the convergence (62) and say 'KS-weakly w.r.t. (63)' resp.'KS-strongly w.r.t. (63)' if we refer to the convergence (63). We finally record a property of KS-weak convergence that will be useful later on.
Proof. For any w ∈ F we have lim m w| X (m) = w KS-strongly by (58). Fix w ∈ F . Clearly sup m w| X (m) f m L 2 (X (m) ,µ (m) ) < +∞ by the boundedness of w, hence lim k w| X (m k ) f m k = w KS-weakly for some w ∈ L 2 (X, µ) and some sequence (m k ) k . For any v ∈ F we have vw ∈ F and trivially (vw)| X (m) = v| X (m) w| X (m) , hence what by the density of F in L 2 (X, µ) implies w = wf and therefore the lemma.

Boundedness and convergence of vector fields.
We provide a version of Proposition 4.1 for finitely ramified sets. Recall the notation from Assumption 5.5.
(i) Given b ∈ H and M > 0 let n 0 be such that Then for all g ∈ F we have .
Then for all g ∈ F (m) we have Proof. We use the shortcut (g) Xα = 1 µ(Xα) Xα g dµ. For any α ∈ A n0 and x ∈ X R ∩ X α have, by (1), Creating a finite partition of X R from the cells X α , α ∈ A n0 , we see that the preceding estimate holds for all x ∈ X R , and using (10) we obtain (i). Statement (ii) is similar.
Similarly as in Corollary 4.1, uniform norm bounds on the vector fields allow to choose uniform constants in the Hardy condition (20). .

Setting (66)
we obtain the desired result by Proposition 5.1.
To formulate an analog of Corollary 4.2 for varying spaces we need a certain compatibility of the vector fields involved. One rather easy way to ensure the latter is to focus on suitable elements of the module Ω 1 a (X R ) and their equivalence classes in H and H (m) which then define vector fields b on X and b (m) on X (m) suitable to allow an approximation procedure. Given an element of Ω 1 a (X R ) of the special form i g i · d u f i with g i ∈ C(X R ) and f i ∈ H n (X), let b defined as its H-equivalence class i g i · d u f i H as in Section 2, that is, By Assumption 5.3 we have f i | X (m) ∈ F (m) for all i and m, so that i g i | X (m) · d u (f i | X (m) ) is an element of Ω 1 a (X (m) ). We define b (m) to be its H (m) -equivalence class The following convergence result may be seen as a partial generalization of (50). It is immediate from Corollary 5.1 and bilinear extension.  (67) and (68) and g ∈ C(X R ). Then we have One might argue that an analog of Corollary 4.2 in terms of a simple restriction of vector fields b ∈ H to X (m) would be more convincing than Corollary 5.4. However, as H and H (m) are obtained by different factorizations, it is not obvious how to correctly define a restriction operation on all of H. Using the finitely ramified cell structure one can introduce restrictions b| X (m) to X (m) of certain types of vector fields b ∈ H and obtain an counterpart of (69) with these restrictions b| X (m) in place of the b (m) 's. This auxiliary result is discussed in Section 7, it is not needed for our main results. Let γ M be as in (66) andγ M similarly but with theb (m) in place of b (m) and suppose that c ∈ C(X R ) satisfies (42). Then for each m the form (Q (m) , F (m) ) as in (60) is a closed form on L 2 (X (m) , µ (m) ), and (25) (26). There is a constant K > 0 such that for each m the generator (L Q (m) , D(L Q (m) )) of (Q (m) , F (m) ) obeys the sector condition (28) with sector constant K. As a consequence, we can observe the following uniform energy bounds on solutions to elliptic and parabolic equations similar to Proposition 4.2.
Proposition 5.2. Let a, b (m) ,b (m) and c be as above such that (70) and (42) hold.
(i) If f ∈ L 2 (X, µ), and u m is the unique weak solution to (29)   (i) If f ∈ L 2 (X, µ), and u m is the unique weak solution to (29) with L Q (m) in place of L and f m = Φ m f in place of f then each subsequence (u m k ) k of (u m ) m has a further subsequence (u m k j ) j such that m k j u m k j ) j converges to a limit u ∈ C(X R ) uniformly on X R . (ii) Ifů ∈ L 2 (X, µ), and u m is the unique solution to (34) in L 2 (X (m) , µ (m) ) with L Q (m) in place of L and with initial conditionů m = Φ mů then for any t > 0 each subsequence (u m k (t)) k of (u m (t)) m has a further subsequence (u m k j (t)) j such that (Ext m k j H (m k j ) m k j u m k j (t)) j converges to a limit u t ∈ C(X R ) uniformly on X R .

5.5.
Generalized strong resolvent convergence. The next result is an analog of Theorem 4.1 on varying spaces, it uses notions of convergence along a sequence of varying Hilbert spaces, [76,97], see Appendix A. The key ingredient is Theorem A.1 -a special case of [98,Theorem 7.15,Corollary 7.16 and Remark 7.17], which constitute a natural generalization of [38,Theorem 3.1] to the framework of varying Hilbert spaces, [76]. A version for more general coefficients is stated below in Theorem 5.2. (67) and for any m let as in (68). Let a ∈ H n (X) be such that (19) holds and let c ∈ C(X R ). Then lim m L Q (m) = L Q in the KS-generalized resolvent sense, and the following hold.
(i) If f ∈ L 2 (X, µ), u is the unique weak solution to (29) on X and u m is the unique weak solution to (29) on X (m) with L Q (m) and Φ m f in place of L Q and f , then we have lim m u m = u KS-strongly.
The proof of Theorem 5.1 makes use of the following key fact.
Lemma 5.3. Suppose (n k ) k is a sequence with n k ↑ +∞ and (u k ) k is a sequence with u k ∈ L 2 (X (n k ) , µ (n k ) ) converging to u ∈ L 2 (X, µ) KS-weakly and satisfying sup k E (n k ) 1 (u k ) < ∞. Then we have u ∈ F , and there is a sequence (k j ) j with k j ↑ +∞ such that (i) lim j u n k j = u KS-weakly w.r.t. (63), and moreover, for any f ∈ F and any sequence (f j ) j such that f j ∈ F (n k j ) and lim j f j = f KS-strongly w.r.t. (63) along (n kj ) j we have (ii) lim j Ext n k j H (n k j ) n k j u n k j = u uniformly on X.
Proof. Let v k := Ext n k H (n k ) n k u k . By hypothesis and (49) we have n k u k , (74), (64) and (53) allow to conclude that (75) lim what implies that lim k v k | X (n k ) = u KS-weakly. We now claim that for any n and any w ∈ H n (X) we have We clearly have lim k Φ n k w = w KS-strongly. Therefore w, u L 2 (X,µ) = lim k Φ n k w, v k | X (n k ) L 2 (X n k ,µ (n k ) ) , and using (58) and (74) this limit is seen to equal Applying (57) we arrive at (76). By (74), and since (59) implies sup k v k L 2 (X,µ) < +∞, we can find a sequence (k j ) j with lim j k j = +∞ such that (u kj ) j converges KS-weakly w.r.t. (63) to a limit u E ∈ F and (v kj ) j converges weakly in L 2 (X, µ) to a limit u E ∈ F . Since n≥0 H n (X) is dense in L 2 (X, µ) we have u E = u by (76), what shows that u ∈ F . We now verify that For any w ∈ H n (X) the equalities hold, the second and third equality due to (57) and (49), respectively. Using (58) twice on the second summands in the last line, the above limit is seen to equal .
For j so large that n kj ≥ n the function w| X (n k j ) is an element of H n k j (X (n k j ) ), so that by orthogonality in n k j u kj in the first summand by u kj . In the second term we can make the same replacement by (53) and (64), so that the above can be rewritten (63). Since n≥0 H n (X) is dense in F , this implies (77) and therefore the first statement of (i), so far for the sequence (u kj ) j . The statement on the limit (73) in (i) follows by Corollary 5.2.
To save notation in the proof of (ii) we now write (u k ) k for the sequence (u kj ) j extracted in (i). Let x 0 ∈ V 0 . Then (1) implies that (v k − v k (x 0 )) k is an equicontinuous and equibounded sequence of functions on X R , so that by Arzelà-Ascoli we can find a subsequence (v kj − v kj (x 0 )) j which converges uniformly on X R to a function w x0 ∈ C(X R ). Since µ is finite, this implies lim j v kj − v kj (x 0 ) = w x0 in L 2 (X, µ). By (58) and (74) we also have ) (x 0 )) = w x0 KS-strongly and therefore also KSweakly. Since lim k v k | X (n k ) = u KS-weakly by (75), we may conclude that lim k v k | X (n k ) (x 0 ) = u − w x0 KS-weakly. In particular, by [76,Lemma 2.3], Since lim m µ (m) (X (m) ) = µ(X) > 0 it follows that v kj | X (n k j ) (x 0 ) is a bounded sequence of real numbers and therefore has a subsequence converging to some limit z ∈ R. Keeping the same notation for this subsequence, we can use (58) and (64) to conclude that lim j v kj | X (n k j ) (x 0 ) = z KS-weakly and therefore necessarily z = u − w x0 . This implies that lim j v kj = lim j (v kj − v kj (x 0 )) + lim j v kj (x 0 ) = u uniformly on X R as stated in (ii). Clearly the statements in (i) remain true for this subsequence.
We prove Theorem 5.1.
Proof. Since the operators L Q (m) obey the sector condition (28) with the same sector constant, Theorem A.1 will imply the desired convergence, provided that the forms Q (m) and Q satisfy the conditions in Definition A.2. Corollary 5.5 then takes care of the claimed uniform convergences.
Without loss of generality we may (and do) assume that the function c ∈ C(X R ) satisfies condition (42). Otherwise we use the same shift argument as in the proof of Theorem 4.1, the statements on uniform convergence then follow using Corollary 5.5.
By (23), (24), (25) and (26) together with Proposition 4.1 and Corollaries 5.3 and 5.4 we can find a constant C > 0 such that for any sufficiently large m we have To check condition (i) in Definition A.2, suppose that (u m ) m is a sequence with u m ∈ L 2 (X (m) , µ (m) ) converging KS-weakly to a function u ∈ L 2 (X, µ) and such that lim m Q (m) 1 (u m ) < +∞. It has a subsequence (u m k ) k which by (78) satisfies sup k E (m k ) (u m k ) < +∞, and by Lemma 5.3 we then know that u ∈ F , what implies the condition.
To verify condition (ii), suppose that u ∈ F , (m k ) k is a sequence with m k ↑ +∞ and that u k ∈ L 2 (X (m k ) , µ (m k ) ) are such that lim k u k = u KS-weakly and sup k Q (u k ) < +∞. Now let w ∈ H n (X). Clearly lim m w| X (m) = w KS-strongly. By Lemma 5.2 we may assume that along (m k ) k we also have lim k a| X (m k ) u k = au and lim k (wĝ i )| X (m k ) u k = wĝ i u KSweakly for all i, otherwise we pass to a suitable subsequence. By (5) also sup k E (m k ) 1 (a| X (m k ) u k ) < +∞ and sup k E (m k ) 1 ((wĝ i )| X (m k ) u k ) < +∞. By Lemma 5.3 we can therefore find a sequence (k j ) j as stated so that (i) and (ii) in Lemma 5.3 hold simultaneously for the sequences (u kj ) j , (a| with limits u, au and wĝ i u, respectively. Our first claim is that To see this note first that by the Leibniz rule for ∂ (m k j ) each element of the sequence on the left hand side equals The first term converges to ∂w, ∂(au) H by (73). In the second summand we can replace u kj by H (m k j ) m k j u kj , note that by (10) and (53) = ∂w, u · ∂a H by Corollary 5.4 and polarization. Using the Leibniz rule for ∂ we arrive at (79). We next claim that Each element of the sequence on the left hand side is a finite linear combination with summands The first term converges to ∂f i , ∂(wĝ i u) H by (73). To see that let ε > 0 and choose n ′ so that by (48) we have and by (54) therefore for large enough j. Since as before we can replace u kj by u| By Corollary 5.4 and (82) we have Since ε was arbitrary, we can combine these two estimates to conclude (81) and therefore (80). The identity follows by linearity from the fact that by Lemma 5.3 (ii) and Corollary 5.4 we have Together with the obvious identity are sectorial closed forms on L 2 (X (m) , µ (m) ), respectively. Moreover, writing (L Q (n,m) , D(L Q (n,m) )) for the generator of the form (Q (n,m) , D(Q (n,m) )), we can observe the following.
(i) If f ∈ L 2 (X, µ), u is the unique weak solution to (29) on X and u (ii) Ifů ∈ L 2 (X, µ), u is the unique solution to (34) on X and u (m) n is the unique weak solution to (34) on X (m) with L Q (n,m) and Φ mů in place of L Q andů, then for any t > 0 there are sequences (m k ) k and (n l ) l with m k ↑ +∞ and n l ↑ +∞ so that Remark 5.7. By [6, Corollary 1.16] we can find a sequence (l k ) k with l k ↑ +∞ such that in the situation of Theorem 5.2 (i) and similarly for (ii).
The following is a straightforward consequence of the density of H * (X) in F , we omit its short proof.
We prove Theorem 5.2.
Proof. Given a ∈ F , let (a n ) n ⊂ H * (X) be a sequence approximating a uniformly on X R and such that all a n satisfy (19) with the same constants 0 < λ < Λ as a. Let M > 0 be large enough such that with f n,i ,f n,i , g n,i ,ĝ n,i ∈ H * (X) that approximate b andb in H, respectively. For each n we can proceed as in (68) and consider the elements of H (m) . With γ M andγ M as in (66) and assuming that, without loss of generality, c ∈ C(X R ) satisfies (42), we can conclude that for each n and each sufficiently large m the forms (Q (n,m) , D(Q (n,m) )) as in (85) with D(Q (n,m) ) = F (m) are closed forms in L 2 (X (m) , µ (m) ).
To prove (i), suppose that f ∈ L 2 (X, µ) and u is the unique weak solution to (29) on X R . Let u = u 1 uniformly on X R . Repeated applications of Theorem 5.1 allow to thin out (m k ) k further so that for any n we have provided that k is greater than some integer k n depending on n. On the other hand Theorem 4.1 allows to find a sequence (n l ) l with n l ↑ +∞ such that lim l→∞ u n l = u uniformly on X, and combining these facts, we obtain (i). Statement (ii) is proved in the same manner.
6. Discrete and metric graph approximations 6.1. Discrete approximations. We describe approximations in terms of discrete Dirichlet forms, our notation follows that of Subsection 5.1. Let (E, F ) be a local regular resistance form on the compact space (X R , R), obtained under Assumption 5.1 as in Section 5.1, and suppose that also Assumption 5.2 is satisfied. Let X (m) = V m , E (m) = E Vm and F (m) = ℓ(V m ) be the discrete energy forms on the finite subsets V m as in (3). Clearly Assumption 5.3 is satisfied, note that for every u ∈ H m (X) we have E Vm (u| Vm ) = E(u) and that (50) is immediate from (12). Since every element of ℓ(V m ) is the pointwise restriction of a function in H m (X), the operator H (m) m is the identity operator id F (m) , so that Assumption 5.4 is trivially satisfied, as pointed out in Remark 5.3. Now let µ be a finite Borel measure on X R such that for any m the value V (m) := inf α∈Am µ(X α ) is strictly positive. Following [86] we define, for each m, a measure µ (m) on V m by where ψ p,m ∈ H m (X) is the (unique) harmonic extension to X of the function 1 {p} on V m . Since X (m) α = V α and p∈Vα ψ p,m (x) = 1 for all m, α ∈ A m and x ∈ X α , we have for all m and α ∈ A m , so that Assumption 5.5 (i) is seen to be satisfied. For each m let Φ m be a linear operator Φ m : In [86, proof of Theorem 1.1] it was shown that for each m the adjoint Φ * m of Φ m equals the harmonic extension operator Ext m : which satisfies Ext m f L 2 (X,µ) ≤ f ℓ 2 (Vm,µ (m) ) for all f ∈ ℓ 2 (V m , µ (m) ). Consequently (55) is fulfilled, and also (59) holds. The function ψ p,m is supported on the union of all X α , α ∈ A m , which contain the point p. By Assumption 5.2 we therefore have (86) lim If a sequence (u m ) m ⊂ F is such that sup m E(u m ) < ∞ then by (1) it is equicontinuous, and combined with (86) it follows that given ε > 0 we have whenever m is large enough, and consequently for such m, note that p∈Vm ψ p,m (x) = 1 for all m and x ∈ X R . This shows (58). For every u ∈ F it follows that since u is bounded and lim m p∈Vm X (u(p) − u(x))ψ p,m (x)dµ(x) = 0 by (86) as above, proving (56). To verify the remaining condition (57) note that for u ∈ H n (X) we have (55) and (58) condition (57) now follows.
Examples 6.1. It is well known that p.c.f. self-similar structures form a subclass of finitely ramified sets. Because of its importance, and since we will discuss metric graph approximations for this subclass in the next section, we provide some details. Let (K, S, {F j } j∈S ) be a connected post-critically finite (p.c.f.) selfsimilar structure, see [63,Definitions 1.3.1,1.3.4 and 1.3.13]. The set of finite words w = w 1 w 2 ...w m of length |w| = m over the alphabet S is denoted by W m := S m , and we write W * = m≥0 W m . Given a word w ∈ W m we write F w = F w1 •F w2 •...•F wm and use the abbreviations K w := F w (K) and V w := F w (V 0 ). Then (K, {K w } w∈W * , {V w } w∈W * ) is a finitely ramified cell structure in the sense of Definition 5.1. We consider the discrete sets V m := ∪ |w|=m V w , m ≥ 0, and assume that ((E Vm , ℓ(V m ))) m is a sequence of Dirichlet forms associated with a regular harmonic structure on K, [63, Definitions 3.1.1 and 3.1.2], that is, there exist constants r j ∈ (0, 1), j ∈ S, a Dirichlet form E V0 (u) = 1 2 p∈V0 q∈V0 c(0; p, q)(u(p) − u(q)) 2 on ℓ(V 0 ), for all m ≥ 1 we have where r w := r w1 . . . r wm for w = w 1 ...w m , and (E Vm+1 ) Vm = E Vm for all m ≥ 0. The regularity of the harmonic structure implies in particular that Ω = K, [63,Theorem 3.3.4], and the limit (3) defines a local regular resistance form (E, F ) on K. Assumptions 5.1 and 5.2 are clear from general theory, [63].
6.2. Metric graph approximations. We describe approximations in terms of local Dirichlet forms on metric graphs (also called 'cable-systems' in [10]). We follow the method in [47] and therefore specify to the case where X is a post-critically finite self-similar set K. Let the setup and notation be as in Examples 6.1.
For each m ≥ 0 we consider V m as the vertex set of a finite simple (unoriented) graph G m = (V m , E m ) with two vertices p, q ∈ V m being the endpoints of the same edge e ∈ E m if there is a word w of length |w| = m such that F −1 w p, F −1 w q ∈ V 0 and c(0; F −1 w , F −1 w q) > 0. For each m and e ∈ E m let l e be a positive number and identify the edge e with an oriented copy of the interval (0, l e ) of length l e , we write i(e) and j(e) for the initial and the terminal vertex of e, respectively. This yields a sequence (Γ m ) m≥0 of metric graphs Γ m , and for each m the set X Γm = V m ∪ e∈Em e, endowed with the natural length metric, becomes a compact metric space See [47] for details and further references. By construction we have X Γm ⊂ X Γm+1 and X Γm ⊂ K for each m.
On the space X Γm we consider the bilinear form (E Γm ,Ẇ 1,2 (X Γm )), where Here f e is the restriction of f to e ∈ E m andẆ 1,2 (e) is the homogeneous Sobolev space consisting of locally Lebesgue integrable functions g on the edge e such that E e (g) := le 0 (g ′ (s)) 2 ds < +∞, where the derivative g ′ of g is understood in the distributional sense. Each form E e , e ∈ E m , satisfies for any f ∈Ẇ 1,2 (X Γm ) and any s, s ′ ∈ e. See [47] for further details. We approximate K, endowed with (E, F ) as in Examples 6.1, by the spaces X (m) = X Γm carrying the resistance forms E (m) = E Γm with domains F (m) =Ẇ 1,2 (X Γm ).
To a function f ∈Ẇ 1,2 (X Γm ) which is linear on each edge e ∈ E m we refer as edge-wise linear function, and we denote the closed linear subspace ofẆ 1,2 (X Γm ) of such functions by EL m . If f ∈ EL m , then its derivative on a fixed edge e is the constant function f ′ e = l −1 e (f (j(e)) − f (i(e))), so that on each e ∈ E m . For a general function f ∈Ẇ 1,2 (X Γm ) formula (89) becomes an inequality in which the left hand side dominates the right hand side. Given a function g ∈ ℓ(V m ) it has a unique extension h to X Γm which is edge-wise linear, h ∈ EL m . In particular, if f ∈ H m (K) is an m-piecewise harmonic function on the p.c.f. self-similar set K then its pointwise restriction f | XΓ m to X Γm is a member of EL m , and Since any such f ∈ H m (K) is uniquely determined by its values on V m ⊂ X Γm , this restriction map is injective, and Assumption 5.3 (i) is seen to be satisfied. Assumption 5.3 (ii) is verified in the following lemma. By ν (m) f we denote the energy measures associated with the form (E Γm ,Ẇ 1,2 (X Γm )). Proof. For f ∈ F and nonnegative g ∈ C(K) we have see [27,Section 3.2]. This implies the relation K g dν f = lim m K g dν Hm(f ) , which by the standard decomposition g = g + − g − remains true for arbitrary g ∈ C(K). For any m we have and given ε > 0 we have sup e∈Em sup s,t∈e |g(s) − g(t)| < ε whenever m is large enough, and in this case, Combining, it follows that lim m K g dν f = lim m K g dν Hm(f )|X Γm .
We verify condition (53) in Assumption 5.4 in the present setup. It states that the small oscillations on the interior of individual edges in X Γm subside uniformly for sequences of functions with a uniform energy bound. Proof. By (88) we have on each e ∈ E m and consequently By H Γm we denote the orthogonal projection inẆ 1,2 (X Γm ) onto EL m . Given f m ∈Ẇ 1,2 (X Γm ) it clearly follows that f m − H Γm f m ∈Ẇ 1,2 (X Γm ), we have (f m − H Γm f m ) | Vm = 0 and E Γm (f m − H Γm f m ) ≤ E Γm (f m ). We verify (54) in Assumption 5.4. Lemma 6.3. Given f, g ∈ H n (X), we have Proof. We first note that for any m ≥ n the functions f e and g e are linear on any fixed e ∈ E m , f e (t) = f e (0) + f ′ e · t and g e (t) = g e (0) + g ′ e · t, t ∈ [0, l e ], with slopes f ′ e ∈ Ê and g ′ e ∈ R, respectively. Therefore E e (f e ) = l e (f ′ e ) 2 for each such e and similarly for the function g. Since (f g) e (t) = f e (t)g e (t) = f e (0)g e (0) + g e (0)f ′ e · t + f e (0)g ′ e · t + f ′ e g ′ e · t 2 and therefore in particular This implies that for any edge e ∈ E m we have Summing up over e ∈ E m and using (90), we see that In what follows let µ be a finite Borel measure on K so that V (m) := inf |w|=m µ(K w ) > 0 for each m. Given an edge e ∈ E m we set (91) ψ e,m (x) : to obtain a function ψ e,m which satisfies We endow the space X Γm with the measure µ (m) := µ Γm which on each individual edge e ∈ E m equals 1 l e K ψ e,m (x)µ(dx) λ 1 | e , here λ 1 denotes the one-dimensional Lebesgue measure. Writing X (m) w for X Γm ∩ K w = V w = e∈Em,e⊂Kw e, we see that so part (i) of Assumption 5.5 is satisfied. The remaining conditions in Assumption 5.5 (ii)-(iv) now follow from results in [47]: If for each m we consider the linear operator Φ m : 14], is how to implement discrete or metric graph approximations for non-finitely ramified compact resistance spaces. If the space is sufficiently structured (such as for instance the Sierpinski carpet, endowed with its standard energy form) then a careful control of resistance metrics should permit to replace several of our arguments based on the cell structure by suitable metric arguments, and it might be possible to replace (86) by certain decay properties of the tent functions. A second follow up open question is how to establish approximations by graph-like manifolds, [87], of non-symmetric forms of type (21) on finitely ramified spaces, and a transparent explanation of how to approximate drift and divergence terms should be quite interesting. A third intriguing open question is how to establish approximations in energy norm. This would most likely have to involve second order splines as for instance discussed in [94] for the case of the Sierpinski gasket endowed with its standard energy form and the self-similar Hausdorff measure. Several tools used in the present paper rely heavily on the use of linear and harmonic functions, and second order version are not so straightforward to see. A fourth natural question to ask, in particular in connection with related problems in probability, [16], is how to approximate equations involving nonlinear first order terms. Although there are results on the convergence of certain non-linear operators along varying spaces, [98], they do not cover these cases.

Restrictions of vector fields
As mentioned in Remark 5.4, a finitely ramified cell structure also permits a restriction operation for specific vector fields. As discussed in [47] the spaces Im ∂ and F / ∼ are isometric as Hilbert spaces, and similarly for Im ∂ (m) and F (m) / ∼. Recall also that for each m the pointwise restriction u → u| X (m) is an isometry from H m (X)/ ∼ onto H m (X (m) )/ ∼. Therefore (67) and (68) give rise to a well defined restriction of gradients of n-harmonic functions: Given f ∈ H n (X) and m ≥ n we can define the restriction of ∂f to X (m) by (93) (∂f )| X (m) := ∂ (m) (f | X (m) ), and this operation is an isometry from ∂(H m (X)) onto ∂ (m) (H m (X (m) )), see for instance [47,Subsection 4.4]. In the sequel we assume, in addition to the assumptions made in Section 5, that for each m and each α ∈ A m the form E α (u) = 1 2 p∈Vα q∈Vα c(m; p, q)(u(p)−u(q)) 2 , u ∈ F , is irreducible on V α . Following [54] we define subspaces H m of H by Proof. Let ε > 0. Choose n g ≥ n sufficiently large such that sup β∈An g sup x,y∈X β |g(x) 2 − g(y) 2 | < ε 5 α∈An E(h α ) .
The energy measures ν hα are nonatomic, hence by (50) and the Portmanteau lemma we can find a positive integer m ε ≥ n g so that for all m ≥ m ε and all α ∈ A n we have Since (99) implies we can use (96) and (97)  Combining (100), (101) and the fact that g · b 2 H = α∈An Xα g 2 dν hα , we arrive at (95).
Appendix A. Generalized strong resolvent convergence The notation in this section is different from that in the main text. We review a special case of the notion of convergence for bilinear forms as studied in [97] (and, among more general results, also in [98]). It covers in particular the case of coercive closed forms, [78]. The results in [97] are generalization of results in [38,Section 3] to the framework of varying Hilbert spaces in [76].
In [  Remark A.2. For any λ > ω the sequence (R Am λ ) m satisfies sup m R Am λ < M (λ − ω) −1 . In the classical case where H m ≡ H and Φ m ≡ id H for all m we therefore observe that the sequence of operators (A m ) m as in (iv) converges to A as in (iv) in the KS-generalized strong resolvent sense if and only if it converges to A in the usual strong resolvent sense, see [58, Section 8.1] (or [88, Section VIII.7] for the self-adjoint case).
One can also introduce a generalization of Mosco convergence for coercive closed forms (not necessarily symmetric). The following definition is a shorted version for coercive closed forms, [78], of [98, Definition 7.14] (see also [97,Definition 2.43]) sufficient for our purposes. We use notation (22) to denote the symmetric part of a bilinear form.
Definition A.2. A sequence ((Q (m) , D(Q (m) ))) m of coercive closed forms (Q (m) , D(Q (m) )) on H m , respectively, with uniformly bounded sector constants, sup m K m < +∞, is said to converge in the KS-generalized Mosco sense to a coercive closed form (Q, D(Q)) on H if there exists a subset C ⊂ D(Q), dense in D(Q), and the following two conditions hold: In [38,97,98] one can find further details. The next Theorem is a special case of [ Theorem A.1. For each m let (Q (m) , D(Q (m) )) be a coercive closed form on H m and assume that the corresponding sector constants are uniformly bounded, sup m K m < +∞. Let G Q (m) α α>0 , T Q (m) t t>0 and (L Q (m) , D(L Q (m) )) be the associated resolvent, semigroup and generator on H m . Suppose that (Q, D(Q)) is a coercive closed form on H with resolvent G Q α α>0 , semigroup T Q t t>0 and generator (L Q , D(L Q )). Then the following are equivalent: (1) The sequence of forms (Q (m) , D(Q (m) )) m converges to (Q, D(Q)) in the KS-generalized Mosco sense.