This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains , where . Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings.

中文翻译：

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具有诺依曼边界条件的扩展 Fisher-Kolmogorov 方程的有限元分析

本文深入研究了开放有界凸域内的扩展 Fisher-Kolmogorov (EFK) 方程的数值分析，其中 。引入了两种不同的有限元方案，即半离散和全离散有限元近似。半离散和全离散有限元近似都建立了解的存在性和唯一性。研究了不同场景下的误差界限，包括半离散解和精确解之间的比较，以及半离散解和全离散解之间以及全离散解和精确解之间的比较。提出了一种有效的算法来求解每个时间步长的完全离散有限元近似产生的非线性系统。该论文还提供了计算的数值误差结果，并展示了各种数值实验，以进一步说明和支持研究结果。