Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data

Abstract

-In this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method (i.e., a standard finite difference operator on piecewiseuniform fitted meshes over the axes x 1 and x 2 ) which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly at the rate ofhere the values of N and N 0 define the number of nodes in the space (with respect to each variable) and time meshes. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.2000 Mathematics Subject Classification: 65N06; 65N22; 35B25.Keywords: singular perturbation, finite difference scheme, fitted mesh, ε-uniform convergence. IntroductionAs is known, depending on the data of a boundary-value problem, in particular, on their smoothness, the solutions of singularly perturbed problems can be sufficiently smooth for each fixed value of the perturbation parameter ε multiplying the highest derivatives. However, the derivatives of solutions grow without bounds (in boundary and transient layers) as the parameter ε tends to zero. This is the reason why numerical methods developed for regular problems (see, e.g. [7,9]) yield errors which depend on an inverse power of the perturbation parameter ε; such errors become large for small ε which is clearly unsatisfactory. Approximation of singularly perturbed equations with non-smooth data 299For certain problems having sufficiently smooth solutions, ε-uniform numerical methods which converge irrespective of ε have been developed and thoroughly studied (see, e.g., [1,10] and the references therein) with the convergence analysis given in the maximum norm. However, conditions imposed on the problem data (precisely, on their smoothness and the order of compatibility conditions on nonsmooth parts of the boundary), which ensures that the solution is sufficiently smooth, are restrictive and, as a rule, are substantially overstated so as to make it difficult to use these methods in practice. For example, in [13] (see also [3,8,10]) for a reaction-diffusion problem a sufficiently high level of smoothness of the coefficients and source terms (from the class C l (G), l > 6) and the boundary functions (from the class C l (S j ) ∩ C(S), l > 6, where S j are the sides forming the boundary S of the set G) was required in o...

Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

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Grid approximation of a singularly perturbed elliptic convection–diffusion equation in an unbounded domain

“…When condition (2.16) is violated and/or the smoothness of the data is not sufficiently high (for example, if a s , b s , c, f ∈ C α (D), ϕ ∈ C 2+α C(Γ j ), ϕ ∈ C(Γ), α ∈ (0, 1), s, j = 1, 2), the technique from [18,23] allows us to establish the ε-uniform convergence of the formal difference scheme (3.2), (3.8) at the rate O(N −ν ), where the convergence order ν = ν(α) is generally small. When condition (2.16) is violated and/or the smoothness of the data is not sufficiently high (for example, if a s , b s , c, f ∈ C α (D), ϕ ∈ C 2+α C(Γ j ), ϕ ∈ C(Γ), α ∈ (0, 1), s, j = 1, 2), the technique from [18,23] allows us to establish the ε-uniform convergence of the formal difference scheme (3.2), (3.8) at the rate O(N −ν ), where the convergence order ν = ν(α) is generally small.…”