Error Estimates of a Linear Approximation Scheme for Nonlinear Diffusion Problems

where u : (0, T ]× Ω→ R is the unknown function, Ω ⊂ R is a bounded domain with a Lipschitz continuous boundary Γ, 0 < T <∞ and ∂νβ(u) denotes outside normal derivative. Function β : R → R is nondecreasing Lipschitz continuous function satisfying (2.1). Function f : (0, T ]×Ω×R→ R is Lipschitz continuous with Lipschitz constant Lf and satisfies (2.2). We denote g(t, x, s) : = γs+φ(t, x), where γ and φ satisfy (2.3). These problems have been of great interest in recent years both from theoretical and numerical point of view. Method based on the so-called nonlinear Chernoff’s formula has been studied by Rogers, Berger and Brezis in [1]. Their linear approximation scheme corresponding to (1.1)–(1.3) reads as follows: