In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the scheme toward the mild solution of the PDE under polynomial growth condition of the nonlinearity. Our convergence rate are obtain for smooth and non-smooth initial data and is similar to that of the autonomous case. Our convergence result for smooth initial data is very important in numerical analysis. For instance, it is one step forward in approximating non-autonomous stochastic partial differential equations by the finite element method. In addition, we provide realistic conditions on the nonlinearity, appropriated to achieve optimal convergence rate without logarithmic reduction by exploiting the smooth properties of the two parameters evolution operator.

中文翻译：

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二阶半线性非自治抛物线偏微分方程有限元逼近的最优误差估计

在这项工作中，我们使用有限元方法研究了二阶非自治半线性抛物线偏微分方程 (PDE) 的数值近似。据我们所知，文献中只研究了线性情况。使用基于依赖于两个参数的演化算子的​​方法，我们获得了该方案在非线性多项式增长条件下对偏微分方程的温和解的误差估计。我们的收敛速度是针对平滑和非平滑初始数据获得的，类似于自主情况下的收敛速度。我们对平滑初始数据的收敛结果在数值分析中非常重要。例如，这是通过有限元方法逼近非自治随机偏微分方程的一步。此外，