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Error estimates of a linear approximation scheme for nonlinear diffusion problems. (English) Zbl 0820.65055

The aim of this paper is to analyze the accuracy of a linear approximation scheme for nonlinear parabolic problems of the type: \[ \begin{alignedat}{2} u_ t(t,x) - \Delta \beta \bigl( u(t,x) \bigr) & = f \biggl( t,x, \beta \bigl( u(t,x) \bigr) \biggr) \quad & (t,x) & \in Q : = (0,T] \times \Omega, \\ \beta \bigl( u(0,x) \bigr) & = \beta \bigl( u_ 0(x) \bigr) \quad & \text{on } & \Omega, \\ - \partial_ \nu \beta \bigl( u(t,x) \bigr) & = \gamma \beta \bigl( u(t,x) \bigr) + \varphi (t,x) \quad & \text{on } & (0,T] \times \Gamma, \;\end{alignedat} \] where \(u : (0,T] \times \Omega \to \mathbb{R}\) is the unknown function, \(\Omega \subset \mathbb{R}^ N\) is a bounded domain with a Lipschitz continuous boundary \(\Gamma\), \(0 < T < \infty\) and \(\partial_ \nu \beta (u)\) denotes the outside normal derivative.
Error estimates of the numerical solution for both strictly increasing and nondecreasing nonlinearities of \(\beta\) are proved.

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
35K55 Nonlinear parabolic equations
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