A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications
A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications
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DOI：
10.1016/j.camwa.2022.08.032
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发表时间：
2021-08
期刊：
影响因子：
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通讯作者：
T. Le;L. Nguyen;H. Tran
中科院分区：
文献类型：
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作者：
T. Le;L. Nguyen;H. Tran
我们提出了一种新的迭代格式来计算一般拟线性椭圆偏微分方程过定边值问题的数值解。其主要思想是利用拟可逆性方法和合适的Carleman权函数对其线性化进行反复求解。由于Carleman权函数的存在，我们可以使用Carleman估计来证明由上述迭代方案生成的序列收敛于期望解。迭代的收敛速度以指数速率快速，而不需要初始的良好猜测。本文应用该方法计算了一类一般拟线性椭圆型方程和一类一阶Hamilton-Jacobi方程的解。给出了数值结果。
We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.