On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal

Abstract

In this paper, we resolve several long-standing issues dealing with optimal pointwisein time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heatequation. In particular, we study the role played by difference quotients (DQs) in obtaining reducedorder model (ROM) error bounds that are optimal with respect to both the time discretizationerror and the ROM discretization error. When the DQs are not used, we prove that both the PODprojection error and the ROM error are suboptimal. When the DQs are used, we prove that both thePOD projection error and the ROM error are optimal. The numerical results for the heat equationsupport the theoretical results.