论文标题
二阶半连接非自治抛物线PDE的有限元近似的最佳误差估计值
Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs
论文作者
论文摘要
在这项工作中，我们使用有限元方法研究了二阶非自治半抛物线偏微分方程（PDE）的数值近似。据我们所知，只有线性案例才在文献中进行研究。使用基于进化算子的方法，取决于两个参数，我们在非线性的多项式生长条件下获得了该方案对PDE轻度溶液的误差估计。我们的收敛速率是用于平滑和非平滑初始数据的，与自主情况相似。我们的平滑初始数据的收敛结果在数值分析中非常重要。例如，通过有限元方法近似非自治的随机部分微分方程是向前迈出的一步。此外，我们通过利用两个参数演化操作员的平滑性能，为实现最佳收敛率提供了现实的条件，以实现最佳收敛速率而不对对数降低。
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.
