Abstract
Error estimates in the L2 norm for the explicit fully discrete numerical finite volume scheme are derived and proved for Perona‐Malik equation. Numerical example is also presented.
Highlights
The following error estimates hold for Perona-Malik weak solution and numerical solution obtained via finite volume method
The method how to obtain the error estimates is similar to one given in [2], but there are some new terms and some terms must be estimated in different way so we describe the whole proof
The following error estimates for Perona-Malik weak solution and numerical solution obtained via finite volume method hold
Summary
In our case we use J(u)(t, ·) = ∇Gσ ∗ u(t, ·) for t fixed, but we can choose any smoothing operator with these properties. It is well known from the regularity theory of such solutions [6] that, due to given properties of the operator J(u), the weak solution of our problem is a function u ∈ L2(I, W 2,2(Ω)) for initial condition u0 ∈ L∞(Ω). It follows from the embedding theory for dimensions d = 2 and d = 3 that u ∈ C(Ω). To obtain our error estimates we need further regularity of the solution, more precisely u ∈ L2(I, W 2,2(Ω)) ∩ L∞(I, W 1,2(Ω)), ∂ttu ∈ L1(I, L1(Ω)) and ∇(∂tu) ∈ L1(I, L1(Ω))
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