In this paper, we present a family of two-grid algorithms for semi-linear parabolic interface problems based on Partially penalized immersed finite element discretizations. Optimal a priori error estimates are derived both in the energy norm and \(L^2\) norm, under the standard piecewise \(H^2\) regularity assumption for the exact solution. For the nonlinear right hand side, we investigate two-grid methods base on Newton method. The efficiency of the two-grid methods is confirmed theoretically and numerically.

中文翻译：

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用于半线性抛物线界面问题的一系列双网格部分惩罚浸入式有限元方法

在本文中，我们提出了一系列基于部分惩罚浸入式有限元离散化的半线性抛物线界面问题的双网格算法。在精确解的标准分段\(H^2\)正则性假设下，在能量范数和\(L^2\)范数中都导出了最佳先验误差估计。对于非线性右侧，我们研究了基于牛顿法的双网格方法。双网格方法的效率在理论上和数值上都得到了证实。