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Dimension splitting for quasilinear parabolic equations. (English) Zbl 1211.65117

In this interesting paper the authors consider the Cauchy problem
\[ \dot u= f(u)= (f_a+ f_b)(u),\quad u(0)= u_0 \]
in a real Hilbert space with \(f_a+ f_b\subseteq f\), under following assumptions: The operators \(f\), \(f_a\), \(f_b\) are densely defined and maximal dissipative and the range \(I-\lambda(f_a+ f_b)\) (\(I\) is the identity operator) is dense for all \(\lambda> 0\).
Then they prove the convergence of splitting schemes i.e. they prove the existence of \(\lim_{n\to\infty} S({t\over n})^n(u)\) for every \(u\) uniformly in \(t\) on bounded intervals for \(S_i(h):= R_{hf_a} R_{hf_b}\) (Lie splitting) (where \(R_{\lambda g}:= (I-\lambda g)^{-1}\)), \(S_s(h):= {1\over 2}(R_{2hf_a}+ R_{2hf_b})\) (sum splitting) and \(S_{pr}(h):= R_{hf_b}(I+ hf_a)R_{hf_a}(I+ hf_b)\) (Peaceman-Rachford splitting).
These abstract results are applied to a quaisilinear parabolic problem with \[ f(u)= \sum^s_{i=1} (D_i a_i(x,D_iu)+ c_i D_i u+ g_i(x))+ \sum^d_{j=s+1}(D_j a_j(x, D_ju)+ c_j D_j u+ q_j(x)) \]
on a bounded domain in \(\mathbb{R}^d\) under homogeneous Dirichlet boundary conditions, also for the degenerate case when the diffusion coefficients may degenerate, for instance for \(a(x,z)= |z|^{p-2}z\).
The above considerations are tested numerically by one example for Lie and Peaceman-Rachford cases.
Reviewer: S. Burys (Kraków)

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65J08 Numerical solutions to abstract evolution equations
35K59 Quasilinear parabolic equations
34G20 Nonlinear differential equations in abstract spaces
35K90 Abstract parabolic equations
47J35 Nonlinear evolution equations
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