延时微分方程有限元法
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摘要
本文研究了一类延时微分方程，即对具有单延时和多延时的线性和非线性问题的有限元方法及其超收敛性，进行了系统的研究。有限元方法的类型，本文研究了连续有限元和间断有限元。
本文主要结果包括以下4方面：
1．利用单元正交逼近校正技巧，研究了具有一个延时项的线性延时微分方程的连续有限元方法及其超收敛性，并推导了其有限元重构导数的强超收敛性。随后推导了多延时线性延时微分方程的连续有限元方法，利用单元正交逼近校正技巧结合数学归纳法证明了它的超收敛性。最后给出了三个例子进行了数值实验，数值验证了上述的理论结果。
2．对于非线性延时微分方程，首先讨论了单延时情形的连续有限元法及其超收敛性，然后推广到多延时情形的连续有限元法及其超收敛性。最后给出了两个例子进行了数值实验，数值验证了上述的理论结果。
3．对于具有单延时项的延时微分方程问题的间断有限元方法，首先讨论线性情形的有限元及其超收敛性，然后将它们推广到非线性情形。最后分别给出了例子进行了线性和非线性问题的数值实验，数值验证了上述的理论结果。
4．最后本文简单地讨论了二阶延时微分方程连续有限元，间断有限元的计算格式。
For a class of delay-differential equations, the linear and nonlinear delay-equation with one-delay term and multi-delay terms, this paper systemic studies the finite element methods and their superconvergence. For the type of the finite element methods, we study the continuous finite element and the discontinuous finite element.
The following is an outline of the main results of the paper:
1. By application of adjustment orthogonal approximate technique in an element, superconvergence of continuous finite elements for linear delay-differential equations with one variabe delay term is studied and ultracon-vergence of its finite element derivative recovery is deduced. Then we derive superconvergence of continuous finite element for the linear delay-differential equations with multi-delay terms by the mathematics induction. The above theoretical results are tested by three numerical examples.
2. For the nonlinear delay-differential equations, firstly discuss super-convergence of continuous finite element methods for the case with one-delay term. Next expand the results to the case with multi-delay terms. Finally the theoretical results are tested by two numerical examples.
3. For the delay-differential equations with one-delay term, first we study superconvergence of discontinuous finite element methods for the linear case. Next we expand the results to the nonlinear case. Finally the theoretical results are tested by two examples of the linear and nonlinear delay-differential equations.
4. Finally for the two order delay-differential equations, we simply discuss the continuous finite element approximative scheme and the discontinuous finite element approximative scheme.
本文主要结果包括以下4方面：
1．利用单元正交逼近校正技巧，研究了具有一个延时项的线性延时微分方程的连续有限元方法及其超收敛性，并推导了其有限元重构导数的强超收敛性。随后推导了多延时线性延时微分方程的连续有限元方法，利用单元正交逼近校正技巧结合数学归纳法证明了它的超收敛性。最后给出了三个例子进行了数值实验，数值验证了上述的理论结果。
2．对于非线性延时微分方程，首先讨论了单延时情形的连续有限元法及其超收敛性，然后推广到多延时情形的连续有限元法及其超收敛性。最后给出了两个例子进行了数值实验，数值验证了上述的理论结果。
3．对于具有单延时项的延时微分方程问题的间断有限元方法，首先讨论线性情形的有限元及其超收敛性，然后将它们推广到非线性情形。最后分别给出了例子进行了线性和非线性问题的数值实验，数值验证了上述的理论结果。
4．最后本文简单地讨论了二阶延时微分方程连续有限元，间断有限元的计算格式。
For a class of delay-differential equations, the linear and nonlinear delay-equation with one-delay term and multi-delay terms, this paper systemic studies the finite element methods and their superconvergence. For the type of the finite element methods, we study the continuous finite element and the discontinuous finite element.
The following is an outline of the main results of the paper:
1. By application of adjustment orthogonal approximate technique in an element, superconvergence of continuous finite elements for linear delay-differential equations with one variabe delay term is studied and ultracon-vergence of its finite element derivative recovery is deduced. Then we derive superconvergence of continuous finite element for the linear delay-differential equations with multi-delay terms by the mathematics induction. The above theoretical results are tested by three numerical examples.
2. For the nonlinear delay-differential equations, firstly discuss super-convergence of continuous finite element methods for the case with one-delay term. Next expand the results to the case with multi-delay terms. Finally the theoretical results are tested by two numerical examples.
3. For the delay-differential equations with one-delay term, first we study superconvergence of discontinuous finite element methods for the linear case. Next we expand the results to the nonlinear case. Finally the theoretical results are tested by two examples of the linear and nonlinear delay-differential equations.
4. Finally for the two order delay-differential equations, we simply discuss the continuous finite element approximative scheme and the discontinuous finite element approximative scheme.
引文
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