奇摄动对流扩散问题的NIPG方法在Shishkin网格上的超逼近性
Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh
摘要：
分析了Shishkin网格上非对称内罚Galerkin（NIPG）方法求解一类奇摄动对流扩散问题的超逼近结果。根据解和方案的特点，提出了一种新的分析方法。更具体地说，Gaussian Lobatto插值和Gaussian Radau插值分别引入层内和层外。通过在不同的网格点上选择特殊的罚参数，进一步建立了能量范数下的几乎k + 1阶超贴近度.这里k是分段多项式的阶。然后，构造了一个简单的后处理算子。特别是，提出了一种新的分析，该算子的稳定性分析。在此基础上，证明了相应的后处理可以使数值解达到更高的精度。最后，在离散能量范数下，可以得到超收敛。这些理论结论可以通过数值计算得到验证。
英文摘要：
In this paper, we analyze the supercloseness result of nonsymmetric interior penalty Galerkin (NIPG) method on Shishkin mesh for a singularly perturbed convection diffusion problem. According to the characteristics of the solution and the scheme, a new analysis is proposed. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. By selecting special penalty parameters at different mesh points, we further establish supercloseness of almost k + 1 order under the energy norm. Here k is the order of piecewise polynomials. Then, a simple post processing operator is constructed. In particular, a new analysis is proposed for the stability analysis of this operator. On the basis of that, we prove that the corresponding post-processing can make the numerical solution achieve higher accuracy. Finally, superconvergence can be derived under the discrete energy norm. These theoretical conclusions can be verified numerically.
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