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.
翻译：我们提出了一个新的迭代计划,用于计算一个通用准线性椭圆形 PDE 的超定边界值问题的数字解决方案。 主要的想法是使用一个合适的 Carleman 重量函数的准逆性方法,反复解决其线性化问题。 Carleman 重量函数的存在,让我们可以使用Carleman 估算法来证明迭代方案产生的序列与理想的解决方案的趋同。迭代法的趋同速度以指数速速度快速,不需要初步的精确猜测。我们用这种方法来计算一些普通准线性椭圆形方程式和一等量的汉密尔顿-贾科比方程式。提出了数值结果。
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