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A new discrete energy technique for multi-step backward difference formulas

The backward differentiation formula (BDF) is a useful family of implicit methods for the numerical integration of stiff differential equations. It is well noticed that the stability and convergence of the $A$-stable BDF1 and BDF2 schemes for parabolic equations can be directly established by using the standard discrete energy analysis. However, such classical analysis technique seems not directly applicable to the BDF-$\mathbf{k}$ schemes for $3\leq \mathbf{k}\leq 5$. To overcome the difficulty, a powerful analysis tool based on the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim., 3:377-423, 1981] was developed by Lubich et al. [IMA J. Numer. Anal., 33:1365-1385, 2013]. In this work, by using the so-called discrete orthogonal convolution kernels technique, we will recover the classical energy analysis so that the stability and convergence of the BDF-$\mathbf{k}$ schemes for $3\leq \mathbf{k}\leq 5$ can be established. One of the theoretical advantages of our analysis technique is that less spacial regularity requirement is needed on the initial data.