Hölder Estimates for the Regular Component of the Solution to a Singularly Perturbed Convection–Diffusion Equation

A Numerical Method for Singularly Perturbed Convection-diffusion Problems Posed on Smooth Domains

“…The remaining data a 1 , a 2 , b are assumed to be sufficiently regular so that u ∈ C 3,γ (Ω) and such that only exponential boundary layers appear near the outflow edges x = 1, y = 1 and a simple corner layer appears in the vicinity of (1,1). This corner layer is induced not by any lack of sufficient compatibility, but by the presence of the singular perturbation parameter.…”

“…Linß [11] examined a class of fitted finite difference operators (arising from a finite volume formulation) on a tensor product of Shishkin meshes. Using the (L ∞ , L 1 ) stability argument developed by Andreev [1], Linß established that for u ∈ C 4 ( Ω), then…”

“…The solution will now have a regular layer near the outflow boundary x = 1, two characteristic layers along the sides y = 0 and y = 1 and, assuming sufficient compatibility at the inflow corners, corner layers in neighbourhoods of the two outflow corners (1, 0), (1,1). An appropriate piecewise uniform Shishkin mesh can be constructed to capture these layers [18].…”

Numerical Results for Singularly Perturbed Convection-Diffusion Problems on an Annulus

“…If f is sufficiently smooth (f ∈ C 1,λ ( Ω)) and satisfies sufficient compatibility at the four corners (see, e.g., [16]) then u ∈ C 3,λ ( Ω) 2 . Assuming additional regularity (f ∈ C 5,λ ( Ω)) and additional compatibility conditions on the four corners [16], the solution u can be decomposed into a sum of a regular component v ∈ C 3,λ ( Ω), and several layer components (all in the space C 3,λ ( Ω)) u(x, y) = (v + w E + w S + w ES + w N + w EN )(x, y);…”

“…1 The differential operator L is inverse monotone in the sense that: for all z ∈ C 0 ( Ω) ∪ C 2 (Ω), if z(x, y) ≥ 0, (x, y) ∈ ∂Ω and Lz(x, y) ≥ 0, (x, y) ∈ Ω then z(x, y) ≥ 0, (x, y) ∈ Ω. 2 The space C γ (D) is the set of all functions that are Hölder continuous of degree γ with respect to the Euclidean norm ∥ • ∥e. The space C k,γ (D) is the set of all functions in C k (D) whose derivatives of order k are Hölder continuous of degree γ.…”