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

Abstract

We approximate the solution operator $(a,f)\mapsto u$ of $-\operatorname{div}(a\nabla u)=f$ 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--\'Emery bound $\mathrm{BE}(-K,\infty)$: an explicit scheme reaches $\dot H^1$-error $\varepsilon$ uniformly over coefficients with bounded carr\'e 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 $\nabla a\cdot\nabla u$ from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.