Boundary-compatible interacting approximations of quasilinear PDEs on bounded domains

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_ε,f_ε) indexed by a scale parameter ε>0, for which the drift A_ε[t,z]z+f_ε[t,z] takes values in an output space Y suitable for discretization. Second, one discretizes this regularized dynamics by a sampling-reconstruction pair (PN,RN) and obtains an interacting ODE on a finite-dimensional state space VNdN. Our main abstract theorem provides a quantitative estimate of the discrepancy y_εN-y between the lifted discrete solution and the exact one, separating the regularization error χ(ε) from the discretization error (1+L_ε)N-γ, where L_ε measures the size of the regularized drift in the output norm. This makes explicit the trade-off between the regularization scale ε, the discretization scale N, and the possible deterioration of L_ε as ε→ 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_ε remains uniformly bounded, leading to algebraic convergence rates in N for quasi-uniform discretizations.