On a Reliable Numerical Method for a Singularly Perturbed Parabolic Reaction-Diffusion Problem in a Doubly Connected Domain

Abstract

In a space-time domain \(\overline{G}=\overline{D} \times [0,T]\), where \(\overline{D}\) is a doubly connected domain in space—a rectangle \(\overline{D}_1\) with a removed circle \(D_2\), we consider the Dirichlet initial–boundary value problem for a singularly perturbed parabolic reaction–diffusion equation. As \(\varepsilon \rightarrow 0\), boundary layers of different types arise in neighborhoods of smooth parts of the lateral boundary and lateral edges. The boundary layers decrease exponentially with distance from the outer and inner lateral boundaries. We discuss an approach for developing a reliable numerical method based on the earlier techniques for simply connected domains. Our aim is to construct an iterative Schwarz method on overlapping subdomains that cover separately the boundary of the parallelepiped or the boundary of the cylinder. It is required that the method converges \(\varepsilon \)-uniformly in the maximum norm as the number of iterations (and the number of mesh points in the case of a difference scheme) grows. We use the Shishkin meshes that condense in the boundary layers and are piecewise uniform along the normal to the smooth parts of the boundaries. To construct meshes near the outer and inner lateral boundaries, it is proposed to use the Cartesian and cylindrical coordinate systems, respectively.