The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data
The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data
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DOI：
10.1007/s40306-023-00500-w
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发表时间：
2022-03
影响因子：
0.5
通讯作者：
L. Nguyen
中科院分区：
文献类型：
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作者：
L. Nguyen
我们提出了一个全局收敛的数值方法来计算解决一般类拟线性偏微分方程的Neumann和Dirichlet边界条件。结合拟可逆方法和一个合适的Carleman权函数，我们定义了一个映射，其不动点是所考虑的偏微分方程的解。为了找到这个不动点，我们定义了一个递归序列与任意初始项使用相同的方式在压缩原理的证明。应用Carleman估计，我们证明了上述序列收敛到所需的解决方案。另一方面，我们还表明，我们的方法提供了可靠的解决方案，即使给定的数据是嘈杂的。数值例子。
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.