椭圆方程约束的最优边界控制问题的非重叠型区域分解迭代方法

Iterative non-overlapping domain decomposition method for optimal boundary control problems governed by elliptic equations

研究了一类椭圆方程约束的最优边界控制问题的数值求解方法。为了避免运用传统数值方法所产生庞大的计算量,我们采用非重叠型区域分解迭代方法。即:将求解区域Ω分解成若干个非重叠子区域,把上述的最优边界控制问题分解成这些子区域上的局部问题,这些局部问题间的内边界条件采用Robin条件。建立了求解这些局部问题的迭代格式,推导证明了迭代格式的收敛性。最后,给出一个数值算例,验证了迭代格式的有效性。

A numerical method for solving optimal boundary control problems governed by elliptic equations is considered. In order to avoid large amounts of calculation produced by traditional numerical methods. An iterative non-overlapping domain decomposition method is established. The whole domain is divided into many non-overlapping subdomains,and the optimal boundary control problem is decomposed into local problems in these subdomains. Robin conditions are used to communicate the local problems on the interfaces between subdomains. The iterative scheme for solving these local problems is studied,and prove the convergence of the scheme is proved. Finally,a numerical example to prove the validity of the scheme is presented.