波动方程的有限元解法

FINITE ELEMENT METHOD OF WAVE EQUATION

在波动方程的经典解法中,初始条件与边界条件都要充分光滑,介质系数也不能出现间断.但是在地震勘探中常需考虑介质系数在区域的不同部分为分块常数的情况,这就导致了具间断系数的波动方程的研究,这类问题是没有经典解的.为此,必须把解的概念加以拓广.相对于经典解来说,这种拓广了的解称为定解问题的广义解.本文遵循偏微分方程的泛函方法关于弱解的定义,定义波动方程混合问题的弱解(弱解也是一种广义解),并直接由弱解引进有限元法.本文最后把一个二阶常微分方程组的柯西问题化为一个Volterra积分方程组,并用迭代法求解.

In the classic solution of wave equation, it is required that the initial and boundary conditions should be smooth enough and any discontinuities are not allowed in medium coefficients. But for seismic exploration, it is often needed to take different medium coeffieients in different areas and in each area, the coefficient must be a constant. There are no classic solutions in such problems. For this reason, the concept of SOLUTION should be extended somehow. Relatively to the classic solution, the extended solution is called "The Generalized Solution of the Defining Problem".In this paper, the weak solution of wave equation with boundary condittions has been defined by the definition of weak solution in the functional analysis of partial differential equations ( Weak solution is also a generalized solution). From the weak solution , the finite element method is directly introduced. At the end of this paper, the Cauchy problem of second order ordinary differential equation is reduced to a system of Volterra integral equations and it is solved by iteration.