Boundary-compatible interacting approximations of quasilinear PDEs on bounded domains

Abstract

We develop a general operator-theoretic route that turns Kato-type quasilinear evolution systems on a Banach scale $(Z,X)$ into finite-dimensional interacting approximations. The construction proceeds in two steps. First, one introduces a regularized family $(A_{\epsilon },f_{\epsilon })$ indexed by a scale parameter $\epsilon >0$, for which the drift $A_{\epsilon }[t,z]z+f_{\epsilon }[t,z]$ takes values in an output space $Y$ suitable for discretization. Second, one discretizes this regularized dynamics by a sampling-reconstruction pair $(P_N,R_N)$ and obtains an interacting ODE on a finite-dimensional state space $V_N\simeq {\mathbb{R}}^{dN}$. Our main abstract theorem provides a quantitative estimate of the discrepancy $y_{\epsilon }^N-y$ between the lifted discrete solution and the exact one, separating the regularization error $\chi (\epsilon )$ from the discretization error $(1+L_{\epsilon })N^{-\gamma }$, where $L_{\epsilon }$ measures the size of the regularized drift in the output norm. This makes explicit the trade-off between the regularization scale $\epsilon$, the discretization scale $N$, and the possible deterioration of $L_{\epsilon }$ as $\epsilon \to 0$. As a running example, we focus on quasilinear PDEs on bounded Lipschitz domains with boundary conditions. We show that Burenkov's variable-step mollifiers provide a boundary-compatible kernelization: they regularize differential operators into explicit integral-interaction operators supported inside the domain and preserve boundary traces of sufficiently regular fields. In this setting one can choose an output space $Y$ for which $L_{\epsilon }$ remains uniformly bounded, leading to algebraic convergence rates in $N$ for quasi-uniform discretizations.