Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

Abstract

The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.

Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator

“…The corresponding estimates then allow us to derive a priori error estimates for sufficiently regular solutions. Results similar to those obtained here have recently been proven in [18] for the ϑ-scheme on a variable time grid, in [16] for the two-step backward differentiation formula (BDF) on an equidistant grid, in [19] for the two-step BDF on a variable time grid if the operator is the derivative of a potential, and can be found in e.g. [40,Ch.…”

Time discretisation of monotone nonlinear evolution problems by the discontinuous Galerkin method

“…8.2] for the backward Euler method. For other time discretisation methods, we have recently been able to prove similar results (see [5,6] for the two-step backward differentiation formula (BDF) on an equidistant grid, [7] for the two-step BDF on a variable time grid, [8] for the ϑ -scheme on a variable time grid, and [9] for a class of stiffly accurate Runge-Kutta methods) although the assumptions on the underlying operator as well as the convergence results themselves differ from method to method. Moreover, in contrast to the techniques used there, the analysis of the discontinuous Galerkin method is indeed more involved and requires additional techniques due to their non-conforming character and the external approximation of the standard solution space for (1.1) by W I .…”

“…We also remark that the consideration of variable time grids as studied here is a prerequisite for any analysis of adaptive methods. It turns out that, in contrast to other methods (see, e.g., [7,8]), there are no severe restrictions on the sequence of variable time grids.…”

Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

“…An example is given by τ n = τ n−1 (1 + cτ n−1 ) for some c > 0. Similar restrictions are also known in the context of the convergence of time discretization methods for nonlinear parabolic problems (see [11,12]). Moreover, a suitable coupling of the maximum time step size and the spatial discretization parameter m is required.…”