变分$p$ -Laplacian算子的有限差分法
A finite difference method for the variational $p$-Laplacian
摘要：
提出了一种新的变分$p$ -Laplace算子的单调有限差分离散方法，\[ Δ_p u=\text{div}(|\nabla u|^{p-2}\nabla u), \]，并给出了求解相关狄利克雷问题的收敛数值格式.所得到的非线性系统求解使用两种不同的方法：一个基于牛顿-拉夫逊和一个显式方法。最后，我们展示了一些数值模拟支持我们的理论结果。据我们所知，这是第一次单调有限差分离散的变分$p$ -拉普拉斯算子，也是第一次，非齐次问题，这个运营商可以处理数值有限差分格式。
英文摘要：
We propose a new monotone finite difference discretization for the variational $p$-Laplace operator, \[ Δ_p u=\text{div}(|\nabla u|^{p-2}\nabla u), \] and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational $p$-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.
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