This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in R d \mathbb{R}^{d} ( d ∈ N d\in\mathbb{N} ). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension. For instance, it provides additional insights into necessary data structures and the complexity of implementations. Throughout, the focus is on an efficient realization using Matlab built-in functions and vectorization. The fast and vectorized Matlab function can be easily implemented in many other vector languages and is provided in Julia, too. The complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement. Numerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.

中文翻译：

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Matlab 中任意空间维度的高效 P1-FEM

本文讨论了连续分段线性函数 (P1-FEM) 的有限元方法在 右 d \mathbb{R}^{d} （ d ε 氮 d\in\mathbb{N} ）。尽管目前对使用高维单纯划分的有限元方法似乎没有很高的实际需求，但研究独立于维度的方法的有效实现有一些优势。例如，它提供了对必要数据结构和实现复杂性的额外见解。在整个过程中，重点是使用有效的实现MATLAB内置函数和矢量化。快速且矢量化MATLAB函数可以很容易地用许多其他矢量语言实现，并且在 Julia 中也提供了。给出了自适应有限元的完整实现，包括组装刚度矩阵、构建载荷矢量、误差估计和自适应网格细化。数值实验强调了我们免费提供的代码的效率，观察到在不超过内存限制时，相对于元素数量，该代码的复杂度略高于线性。