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Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator. (English) Zbl 1172.65026
The paper is concerned with the initial value problem
\[ u^{\prime}+Au=f \quad \text{in }(0,T), \qquad u(0)=u_0, \]
where the nonlinear operator \(A:V \to V^{*}\) acting on a Gelfand triple \(V\subseteq H \subseteq V^{*}\) is supposed to be a monotone and coercive potential operator that fulfills a growth condition. On the variable time grid \(\mathbb{I}: 0=t_0 < t_1< \dots < t_N=T\) \((N \in \mathbb{N})\) with \(\tau_n=t_n-t_{n-1}\) \((n=1,2,\dots,N)\), \(r_n=\frac{\tau_{n}}{\tau_{n-1}}\) \((n=2,3,\dots,N)\), \(\tau_{\max}=\max_{n=1,2,\dots,N}\tau_{n}\), \(r_{\max}=\max(\max_{n=2,3,\dots,N}r_n,1)\) the two-step temporal semidiscretization for the computation of a time discrete solution \(u^{n}\approx u(t_n)\) with an initial implicit Euler step
\[ Du^{n}+Au^{n}=f^{n}, \quad n=1,2,\dots,N \]
is considered, where
\[ Du^{1}=\frac{1}{\tau_1}\left(u^{1}-u^{0}\right),\quad Du^{n}=\frac{1}{\tau_1}\left(\frac{1+2r_n}{1+r_n}u^{n}-(1+r_n)u^{n-1}+ \frac{r_n^2}{1+r_n}u^{n-2}\right). \]
It is proven that a piecewise linear prolongation of the discrete solution converges to the weak solution of the initial problem provided that the ratios of adjacent step sizes are close to 1 and do not vary too much.

MSC:
65J15 Numerical solutions to equations with nonlinear operators
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
34G20 Nonlinear differential equations in abstract spaces
35K90 Abstract parabolic equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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