Least-squares finite element method for ordinary differential equations
Least-squares finite element method for ordinary differential equations
复制标题
常微分方程的最小二乘有限元法
DOI：
10.1016/j.cam.2022.114660
复制
发表时间：
2023
影响因子：
2.4
通讯作者：
Liu, Honghu
中科院分区：
文献类型：
--
作者：
Chung, Matthias;Krueger, Justin;Liu, Honghu
考虑非线性常微分方程组的最小二乘有限元方法，并建立了该方法在使用分段线性元时的最优误差估计。的主要假设是，向量场是足够光滑的，以及当地的Lipschitz常数以及与非线性相关的雅可比矩阵的运营商范数是足够小的，当限制到一个合适的邻域的真解所考虑的初始值问题。这种理论上的最优性进一步说明了数字，沿着与证据可能扩展到高阶的基础元素。最后通过算例说明了该方法在各种情况下与有限差分法相比的优越性.适当的修改自适应时间步进进行了讨论。
We consider the least-squares finite element method (lsfem) for systems of nonlinear ordinary differential equations, and establish an optimal error estimate for this method when piecewise linear elements are used. The main assumptions are that the vector field is sufficiently smooth and that the local Lipschitz constant as well as the operator norm of the Jacobian matrix associated with the nonlinearity are sufficiently small, when restricted to a suitable neighborhood of the true solution for the considered initial value problem. This theoretic optimality is further illustrated numerically, along with evidence of possible extension to higher-order basis elements. Examples are also presented to show the advantages oflsfemcompared with finite difference methods in various scenarios. Suitable modifications for adaptive time-stepping are discussed as well.