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In this paper, we resolve several long standing issues dealing with optimalpointwise in time error bounds for proper orthogonal decomposition (POD)reduced order modeling of the heat equation. In particular, we study the roleplayed by difference quotients (DQs) in obtaining reduced order 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 provethat both the ROM projection error and the ROM error are suboptimal. When theDQs are used, we prove that both the ROM projection error and the ROM error areoptimal. The numerical results for the heat equation support the theoreticalresults.

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