有限元方法求解椭圆型偏微分方程数值解的可行性分析
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摘要
非线性微分方程近几年发展获得众多领域关注,它涉及经济学、物理学及工程学等学科问题数学模型,文中提出运用有限元方法对椭圆型偏微分方程进行求解,分析方程数值解存在可行性。采用弱有限元思想在问题区域上将其划分为多边形或多面体,使多边体逼近函数中含有间断多项式函数,令单元边界多项式表述单元间关系;同时对间断函数引进广义弱微分算子,应用至变分形式中,使逼近数值解通过稳定子产生弱连续性。基于解弱连续性,利用节点增量方法,对偏微分方程问题区域再次实行三角形单元分划,获得符合Delaunay条件的三角形单元,将单元所有节点进行编号,计算单元上系数矩阵及组装单元矩阵,获知单元节点关系,从而求得椭圆型偏微分方程可行性数值解。
The nonlinear differential equations in recent years many areas of concern,a mathematical model which relates to economics,physics and engineering problems,put forward to use the finite element method for solving elliptic partial differential equations in this paper,numerical solution of the equation has the feasibility analysis. Using the finite element method in the weak region partition divided into arbitrary polyhedron or multilateral body, the multilateral body approximation space contains discontinuous polynomial function,through the unit from the unit boundary polynomial of discontinuous contact; generalized weak differential operator,applied to variational form,the approximation solution by generating stable data weak continuity. Weak continuity of solutions based on the use of node increment method,the partial differential equation problem again the implementation of regional triangle division,obtain the triangular element Delaunay meets the conditions,the unit number of all nodes,calculating unit coefficient matrix and assembling unit matrix,obtaining unit node relationship,thus obtains the feasibility of numerical solution of elliptic partial differential equations.
The nonlinear differential equations in recent years many areas of concern,a mathematical model which relates to economics,physics and engineering problems,put forward to use the finite element method for solving elliptic partial differential equations in this paper,numerical solution of the equation has the feasibility analysis. Using the finite element method in the weak region partition divided into arbitrary polyhedron or multilateral body, the multilateral body approximation space contains discontinuous polynomial function,through the unit from the unit boundary polynomial of discontinuous contact; generalized weak differential operator,applied to variational form,the approximation solution by generating stable data weak continuity. Weak continuity of solutions based on the use of node increment method,the partial differential equation problem again the implementation of regional triangle division,obtain the triangular element Delaunay meets the conditions,the unit number of all nodes,calculating unit coefficient matrix and assembling unit matrix,obtaining unit node relationship,thus obtains the feasibility of numerical solution of elliptic partial differential equations.
引文
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[2]刘旖.分布控制偏微分方程约束下微分代数方程组多目标优化[J].科技通报,2017,33(4):11-14.
[3]邹长军,尹勇,李海江,等.偏微分方程数值计算在虚拟现实中应用与研究[J].舰船科学技术,2017,39(15):164-169.
[4]胡亮,罗懋康.柱面非线性麦克斯韦方程组的行波解[J].物理学报,2017,66(13):22-27.
[5]辜旭赞.论Rossby波在东、西风带中传播与频散II——球面Rossby波方程与诊断分析[J].热带气象学报,2016,32(5):622-633.
[6]吴娇,韩卫卫.一类Wick型随机偏微分方程的白噪声泛函解[J].内蒙古师大学报(自然汉文版),2016,45(2):159-165.
[7]林庆泽.复合函数高阶微商公式的推广[J].忻州师范学院学报,2017,33(2):8-13.
[8]徐鸿鹏,尹社会,张勇.一个超混沌类Lorenz系统的非线性动力学行为及计算机仿真[J].电子设计工程,2016,24(10):38-41.
[9]刘健,赵增勤.Cerami条件下脉冲边值问题古典解的存在性[J].数学学报,2016,59(5):609-622.
[10]胡传峰,姬秀.一类偏微分方程的解的Bernstein性质[J].中北大学学报(自然科学版),2016,37(3):220-224.