Nonlocal Venttsel' diffusion in fractal‐type domains: Regularity results and numerical approximation

Abstract

We study a nonlocal Venttsel' problem in a nonconvex bounded domain with a Koch‐type boundary. Regularity results of the strict solution are proved in weighted Sobolev spaces. The numerical approximation of the problem is carried out, and optimal a priori error estimates are obtained.

Singular p-homogenization for highly conductive fractal layers

“…Mathematically, Venttsel' problems are characterized by an unusual boundary condition: the operators governing the diffusion in the bulk and on the boundary are of the same order. For the literature on Venttsel' problems in piecewise smooth or fractal domains, from linear to quasi-linear, from local to nonlocal, we refer to [34,15,17,31,12,32,14,13]. In the linear smooth case, it is by now well known that Venttsel' problems can be seen as the limit of suitable homogenization problems.…”

Approximation of 3D Stokes Flows in Fractal Domains

“…We point out that the irregular nature of fractal-type boundaries does not allow us to use the standard techniques because the presence of irregular geometries deteriorates the regularity of the solution. For the numerical approximation of boundary value problems in fractal domains we refer to [5,7,4]. More precisely, for every n ∈ N, we consider parabolic incompressible Stokes problems (P n ) in a cylindrical domain Q n with a Koch-type cross section (see Section 1), with homogeneous Dirichlet boundary conditions:…”

Regularity results for nonlocal evolution Venttsel' problems

“…Among the others, we refer to [21], [29], [32], [22] and the references listed in. In this paper, we consider a nonlocal term which can be regarded as a suitable version of the regional fractional Laplacian (−∆) s , for s ∈ (0, 1), see [7] for applications.…”

“…When considering the numerical approximation of this problem, to prove regularity results is a key issue for obtaining optimal a priori error estimates. To this regard, see [8,9] for the local case, and [7] for the nonlocal case, under stronger assumptions on the data.…”