Connections between finite difference and finite element approximations
Connections between finite difference and finite element approximations
复制标题
有限差分和有限元近似之间的联系
DOI：
10.1080/00036811.2021.2005785
复制
发表时间：
2021
影响因子：
1.1
通讯作者：
Cristina Bacuta
中科院分区：
文献类型：
--
作者：
Constantin Bacuta;Cristina Bacuta
我们提出了模型边值问题的有限差分法和有限元法之间的有用联系。我们从观察开始，在有限元环境中，一维解的插值与解的有限元近似相吻合。这个结果可以看作是格林函数公式在连续水平解的推广。我们写出了有限差分和有限元系统，使两个相应的线性系统具有相同的刚度矩阵，并比较了两种方法的右侧载荷向量。利用格林函数的求值，将刚度矩阵的逆公式推广到网格点非均匀分布的情况。根据两种方法之间的联系进行了误差分析，并估计了两种方法之差的能量范数。提供了对2D情况的有趣扩展。
We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.