×

Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations. (English) Zbl 1082.65092

A nonhomogeneous linear parabolic equation of the type \[ u_t+Au =f, \quad \text{for } t>0, u(0)=\nu \] in a Hilbert space \(H\) with norm \(\| .\| \) is investigated from a numerically point of view. Here the operator \(A\) is a linear, selfadjoint, positive definite, not necessary bounded operator with compact inverse, \(\nu \in H\), \(f\) is a function of \(t\) with values in \(H\). A multistep backward difference method is used to obtain a numerical solution \(U^n\) which is the approximate solution of \(u(t_n)\) for \(t_n= n\cdot k \), where \(k\) is the time step: \[ \overline \partial_p U^n+AU^n =f^n, \text{ for } n\geq p, \text{ and } f^n=f(t_n), \] where \[ \overline \partial_p U^n = k^{-1} \sum _{\nu =0}^p c_\nu U^{n-\nu}, \] where the coefficients \(c_\nu\) are independent of \(k\). The theory of stability and error estimates for the approximation of the solution by a multistep method for constant as well as variable time steps are well known. The aim of this paper is to consider the smoothing property in the multistep backward difference method and time derivative approximation of this equation. First the smoothing property of the type \[ \| \overline \partial_p U^n\| \leq C t_n^{-1} \sum _{j=0}^{p-1} \| U^j\| , \] for \(p\leq 6\) and \( n\geq 2p\) is proved. Then the error estimates for the approximation \(\overline \partial_p U^n\) of the time derivative \(u_t(t_n)\) for nonsmooth data is derived.

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35A35 Theoretical approximation in context of PDEs
35K65 Degenerate parabolic equations
35K90 Abstract parabolic equations

Keywords:

nonsmooth data
PDFBibTeX XMLCite
Full Text: DOI EuDML