椭圆边值问题与拟共形映射的数值解

本文研究多连通区域上一阶线性椭圆型复方程组的黎曼-希尔伯特边值问题的数值解法,文中提出了与上述边值问题等价的一种变分问题,然后用有限元法求出这种变分问题的近似解,这也是原边值问题的数值解.Klabukova 曾用交分差分方法讨论了广义解析函数上述边值问题的近似解法,由于她使用的方法与共轭方程有关,因此难以将所得结果推广到一般的一阶线性一致椭圆型复方程的情形.在作者过去的工作中,给出了多连通区域上以上边值问题的一种适定提法,由于这种提法不与共轭方程直接相关,因此才有可能将所考虑的边值问题数值求解推进到本文中所述较一般的多个末知函数的一阶椭圆组上去,这种复方程组的解包含广义超解析函数作为特殊情形.作为上述结果的应用,本文还讨论了某些线性拟共形映射的数值求解。

Numerical Solutions of Elliptic Boundary Value Problems and Quasiconformal Mappings

This paper discusses the numerial solutions of Riemann-Hilbert boundary valueproblem for linear elliptic of first order complex equations on multiple connecteddomain.A case of variational problem that is equivalent to the above problem isgiven.By using finite element method,we find out the approximate solution ofthis variational problem,namely,the numerial solution of the original boundaryvalue problem.L.S.Klabukova had discussed the approximate solution of the aboveboundary value problem for generalized analytic functions by using variationalmethod.Due to her modified boundary problem which is concerned with theconjugate equations,her results are difficult to be generalized to the case of generallinear uniformly elliptic complex equations.In my past work,the well posed ofabove boundary value problem on multiple connected domain was given,because thiswell posed is independent of the conjugate complex equation,the numerial solutionmethod is available for a first order elliptic system(1.1),this solution includes thegeneral super analytic functions as a special case.In the end of this paper,thenumerial solutions of some linear equasicomformal maping are discussed