Bessel $(p,s)$-Laplacian 的有限元逼近

中文摘要 AI 辅助

我们研究了Bessel $(p,s)$-Laplacian算子$\text{div}^{s} \big(|\nabla^{s} u|^{p-2}\nabla^{s} u\big)$的Dirichlet问题的有限元逼近。该算子由Riesz分数阶梯度$\nabla^s$构造，并定义在Bessel位势空间$X^{s,p}$上，该空间通过$L^p$与$W^{1,p}$之间的复插值得到。当$p=2$时，该算子退化为分数阶Laplacian；而当$p\neq 2$时，它不同于由实插值得到的分数阶$p$-Laplacian。由于直接的Galerkin离散化需要在每次非线性迭代中重新组装稠密刚度矩阵，我们提出了一种基于投影分数阶梯度$B_h = \boldsymbol{\Pi}_h \nabla^s$的增广拉格朗日公式，其中唯一的稠密矩阵仅组装一次，非线性项在单元级别解耦。我们分析了由此产生的一致性误差，在增广拉格朗日格式的离散inf-sup条件下，建立了$X^{s,p}$中的先验收敛速率，并给出了数值实验，展示了该方法的收敛性及其处理退化和强非线性问题的能力。

英文摘要

We study the finite element approximation of the Dirichlet problem for the Bessel $(p,s)$-Laplacian, $\operatorname{div}^{s} \left(|\nabla^{s} u|^{p-2}\nabla^{s} u\right)$, built from the Riesz fractional gradient $\nabla^s$ and posed on the Bessel potential space $X^{s,p}$, obtained by complex interpolation between $L^p$ and $W^{1,p}$. For $p = 2$ this operator reduces to the fractional Laplacian, while for $p \neq 2$ it differs from the fractional $p$-Laplacian arising from real interpolation. Since a direct Galerkin discretization requires reassembling a dense stiffness matrix at every nonlinear iteration, we propose an augmented Lagrangian formulation based on the projected fractional gradient $B_h = \mathbfΠ_h \nabla^s$, in which the only dense matrix is assembled once and the nonlinearity decouples elementwise. We analyze the resulting consistency error, establish a priori convergence rates in $X^{s,p}$, conditional on a discrete inf-sup condition for the augmented Lagrangian scheme, and present numerical experiments that illustrate the convergence of the method and its ability to deal with degenerate and strongly nonlinear problems.

发表机构

• PEDECIBA – Program for the Development of Basic Sciences and Instituto de Matematica y Estadística “Rafael Laguardia”, Universidad de la República（基础科学发展计划（PEDECIBA）与拉斐尔·拉瓜迪亚数学与统计研究所，乌拉圭共和国大学）

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