双曲型积微分方程动边界问题有限元格式及分析

The Finite Element Approximations and Their Analysis for the Hyperbolic Integro Differential Equation in a Time Dependent Domain

研究具有变动边界的一维区域上的双曲型积微分方程ｕｔ＝ａ（ｘ，ｔ）ｕ（ｘ，ｔ）ｘ＋∫ｔ０ｂ（ｔ，τ，ｘ）ｕ（ｘ，τ）ｘｄτｘ＋ｐ（ｘ，ｔ）ｕ（ｘ，ｔ）ｘ＋ｆ（ｘ，ｔ）ｘ∈Ω（ｔ），ｔ∈Ｊ的初边值问题．提出一类半离散和全离散有限元逼近格式，并表明了后者的稳定性．通过空间变量代换．把问题化成了更易于处理的，定义在固定空间区域上的等价“标准”形式；通过引入Ｒｉｔｚ－Ｖｏｌｔｅｒｒａ投影，有效处理了时间积分项产生的影响；并结合使用其他微分方程先验误差估计技巧，对两格式都得到了最优阶的Ｌ２模和Ｈ１模收敛结果．

In this paper, we study the one dimensional integro differential equation with initial boundary conditions in a time dependent domain for hyperbolic type u tt =a(x,t)u(x,t) x+∫ t 0b(t,τ,x)u(x,t) xdτ x+p(x,t)u(x,t) x+f(x,t) x∈Ω(t),t∈J Both continuous time and discrete time finite element approximations are suggested, the stability of the latter is shown. By changing space variable, we translate the problem into a more disposable, equivalent “standand” form defined in a boundary settled domain; by introducing Ritz Volterra projection, we effectively treat the influence coming from the time integral term; other priori error estimate technique for differential equations is also applied, finally, the optimal L 2 norm and H 1 norm convergence results for the two approximations are derived.