论文标题
带有过度确定边界数据的准线性椭圆方程的Carleman收缩映射方法
The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data
论文作者
论文摘要
我们提出了一种全局收敛的数值方法，以将解决方案计算为具有Neumann和Dirichlet边界条件的一般类线性PDE。结合了准可逆性方法和合适的卡尔曼重量函数，我们定义了一个固定点的图是考虑到正在考虑的PDE的解决方案。为了找到此固定点，我们使用与收缩原理证明相同的方式定义了一个任意初始项的递归序列。应用卡尔曼估计值，我们表明上面的序列会收敛到所需的解决方案。另一方面，我们还表明，即使给定的数据嘈杂，我们的方法也可以提供可靠的解决方案。提出了数值示例。
We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented.
