Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products

We approximate the solution operator $(a,f)\mapsto u$ of $-\mathrm{div}(a\mathrm{\nabla }u)=f$ by operator networks whose only primitives are heat-semigroup functions of a fixed reference operator and pointwise products. The main result is an emulation theorem on strongly local Dirichlet spaces with a spectral gap and a Bakry–Émery bound $\mathrm{B}\mathrm{E}(-K,\mathrm{\infty })$: an explicit scheme reaches ${\dot{H}}^1$-error $\epsilon$ uniformly over coefficients with bounded carré du champ and bounded generator image and over $L^2$ data, with parameters depending only on the coefficient bounds, the accuracy and $K$; on the torus this gives a $C^2$-type coefficient class with constants independent of the dimension. The torus version refines this to Wiener classes of fractional regularity $A^{1+\theta }$, with an explicit rate attained by the scheme’s product-defect block. The scheme unrolls the Laplacian-preconditioned Richardson iteration, controls each inexact step in the operator norm of ${\dot{H}}^1$ without Galerkin truncation, and synthesises the coupling $\mathrm{\nabla }a\cdot \mathrm{\nabla }u$ from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.