Optimal Local Approximation Spaces for Parabolic Problems

We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed multiscale domain decomposition methods. The diffusion coefficient can arbitrarily rough space time. To construct we consider a compact transfer operator acts on solutions covers full time dimension. then given by left singular vectors operator. prove compactness latter combine suitable Caccioppoli inequality with theorem Aubin--Lions. In contrast to elliptic setting [I. Babuška R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406] need an additional regularity result two results. Furthermore, employ generalized finite element method couple global solution. Since our approach yields reduced bases, computation does not require stepping is thus computationally efficient. Moreover, derive rigorous priori error bounds. detail, bound graph norm errors $L^2(H^1)$-norm, noting maps equipped this norm. Numerical experiments demonstrate exponential decay values high or structure regarding