Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem
Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem
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二维奇异摄动半线性反应扩散问题的数值分析
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发表时间：
2009
期刊：
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通讯作者：
N. Kopteva
中科院分区：
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作者：
N. Kopteva
在光滑的二维域中考虑具有多个解的半线性反应扩散方程。其扩散参数e 2 任意小，从而产生边界层。我们扩展了论文[N.科普特娃：数学。比较。 76 (2007) 631–646]，其中假设边界 $partialOmega$ 的参数化是已知的，对于更实际的情况，当域由一组有序的边界点定义时。结果表明，使用层自适应网格，可以在离散最大范数下获得二阶收敛，在 e 中一致，对于 e ≤ Ch。这里h > 0是网格单元的最大边长，而网格节点的数量不超过Ch− 2。给出的数值结果支持我们的理论误差估计。
A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter e 2 is arbitrarily small, which induces boundary layers. We extend the numerical method and its maximum norm error analysis of the paper [N. Kopteva: Math. Comp. 76 (2007) 631–646], in which a parametrization of the boundary $partialOmega$ is assumed to be known, to a more practical case when the domain is defined by an ordered set of boundary points. It is shown that, using layer-adapted meshes, one gets second-order convergence in the discrete maximum norm, uniformly in e for e ≤ Ch. Here h > 0 is the maximum side length of mesh elements, while the number of mesh nodes does not exceed Ch − 2. Numerical results are presented that support our theoretical error estimates.