一类高维非线性色散耗散波动方程的有限元分析

非线性色散耗散波动方程 ,可以用来研究非线性弹性杆中纵向形变波传播及弱非线性作用下空间变换离子声波传播问题 .有限元法是现代数值分析求解各类偏微分方程的重要方法之一 .它具有网格剖分灵活 ,适用区域广泛 ,精度高等特点 .对一类高维非线性色散耗散波动方程 ,运用有限元数值分析方法 ,给出了问题的变分形式和有限元解空间 ,构造了半离散有限元格式和非线性全离散有限元格式 .证明了这两个有限元格式解的存在唯一性 .特别是对非线性全离散有限元格式 ,为了能运用Brouwer不动点原理和压缩映射原理 ,定义并证明了一个压缩映射 .最后 ,利用椭圆投影 ,对这两个格式进行了误差分析 ,得到了有限元解与原方程精确解间的最优L2 模和H1模误差估计

The finite element method for a class of multi-dimensional nonlinear dispersive-dissipative wave equations

Multi-dimensional nonlinear dispersive-diss ipative wave equations are used to represent the propagation problems of lengthways-wave in nonlinear Elas t ic rods and ion-sonic of space transformation by weak nonlinear effect. The fin i te element method is one of important modern numerical methods for solving various partial differential equations. It has many advantages: flexible for mes hing, suitable for many kinds of domain, higher accuracy of computation. The fi nit e element method is used for a class of multi-dimensional nonlinear dispersive -d issipative wave equations. Weak formulation and finite element space are given. Two finite element schemes are formulated: one is semi-discrete scheme; the oth e r is full-discrete nonlinear scheme. The existence and uniqueness of the soluti o n for these two schemes are proved. Especially for full-discrete nonlinear sch e me, in order to use Brouwer's Fixed-Point Theorem and Compressible Mapping Theo r em, a mapping is defined and proved to be compressible. Finally, error analyses of these two schemes are shown by an elliptic projection. And the optimal L 2-norm and H 1-norm error estimate are derived.