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首页> 外文会议>Finite difference methods: theory and applications >On a Reliable Numerical Method for a Singularly Perturbed Parabolic React ion-Diffusion Problem in a Doubly Connected Domain
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On a Reliable Numerical Method for a Singularly Perturbed Parabolic React ion-Diffusion Problem in a Doubly Connected Domain

机译：关于双连通域奇摄动抛物线反应离子扩散问题的可靠数值方法

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摘要

In a space-time domain G = D × [0, T], where D is a doubly connected domain in space-a rectangle D_1with a removed circle D_2, we consider the Dirichlet initial-boundary value problem for a singularly perturbed parabolic reaction-diffusion equation. As ε→ 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 ε-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.
机译：在时空域G = D×[0，T]中，D是空间中的双连通域-具有去除的圆D_2的矩形D_1，我们考虑奇摄动抛物线反应的Dirichlet初边值问题-扩散方程。当ε→0时，在横向边界和横向边缘的平滑部分附近会出现不同类型的边界层。边界层随着与外部和内部横向边界的距离呈指数减小。我们讨论一种基于简单连接域的早期技术开发可靠的数值方法的方法。我们的目的是在重叠子域上构造一个迭代Schwarz方法，该子域分别覆盖平行六面体的边界或圆柱的边界。要求随着迭代次数（以及差分方案情况下的网格点数）的增长，该方法在最大范数上均匀收敛ε。我们使用Shishkin网格，这些网格在边界层中凝聚，并且沿着边界的平滑部分的法线是分段均匀的。为了在外部和内部横向边界附近构造网格，建议分别使用笛卡尔坐标系和圆柱坐标系。

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