[论文解读] FINITE ELEMENT ERROR ESTIMATES FOR CRITICAL EXPONENT SEMILINEAR PROBLEMS WITHOUT ANGLE CONDITIONS

Randolph E. Bank, Michael Holst+2 more|arXiv (Cornell University)|Aug 18, 2011

Advanced Numerical Methods in Computational Mathematics参考文献 38被引用 7

一句话总结

本文为在2D和3D中应用有限元方法求解具有临界指数的半线性问题建立了先验误差估计，且无需对网格施加限制性角度条件。通过用非线性项的局部Lipschitz条件替代基于离散最大值原理的逐点控制，作者导出了拟最优误差界，并表明离散解的L¹控制可由标准有限元逼近理论得出，从而无需依赖几何网格约束。

ABSTRACT

In this article we consider a priori error estimates for semilinear problems with critical and subcritical polynomial nonlinearity in d space dimensions. When d = 2 and d = 3, it is well-understood how mesh geometry impacts finite element interpolant quality. However, much more restrictive conditions on angles are needed to derive ba- sic a priori quasi-optimal error estimates as well as a priori pointwise estimates for Galerkin approximations. These angle conditions, which are particularly difficult to sat- isfy in three dimensions in any type of unstructured or adaptive setting, are needed in order to gain pointwise control of the nonlinearity through discrete maximum princi- ples. In this article, we show how to derive these types of a priori estimates without requiring the discrete maximum principle, hence eliminating the need for restrictive an- gle conditions. We first describe a class of semilinear problems with critical exponents, and highlight some particularly important motivating examples in geometric analysis and general relativity. The solution theory for this class of problems is then reviewed, including generalized maximum principles and the construction of a priori L 1 bounds using cutoff functions and the De Giorgi iterative method (or Stampacchia truncation method). We then develop a basic quasi-optimal a priori error estimate for Galerkin approximations, where the nonlinearity is controlled using only a local Lipschitz prop- erty rather than through pointwise control of the discrete solution, which eliminates the requirement that the discrete solution satisfy a discrete form of the maximum principle. We show that the local Lipschitz property holds for nonlinearities up to and including the critical exponent. We then use some well-known results in finite element approximation theory to show that, under some additional minimal smoothness assumptions, that the a priori error estimate is itself enough to give L 1 control the discrete solution, with- out the need for restrictive angle conditions that would be required to obtain a discrete maximum principle.

研究动机与目标

• 解决现有有限元误差估计在2D和3D中需要限制性网格角度条件的局限性。
• 在推导Galerkin逼近的先验误差界时，消除对离散最大值原理的依赖。
• 仅利用非线性项的局部Lipschitz性质，为具有临界和次临界非线性的半线性问题建立拟最优误差估计。
• 表明离散解的L¹控制可在无需依赖角度的假设下实现，仅依赖标准有限元逼近理论。
• 将有限元方法的适用范围扩展至具有临界非线性的几何和相对论型PDE，在非结构化或自适应网格中保持有效性。

提出的方法

• 作者分析了一类源自几何分析和广义相对论的具有临界指数的半线性问题。
• 他们采用广义最大值原理和De Giorgi迭代方法（或Stampacchia截断）来构造精确解的先验L¹有界性。
• 不强制实施离散最大值原理，而是通过在解上施加非线性项的局部Lipschitz条件来控制其行为，该条件在临界指数范围内依然成立。
• 该方法依赖标准有限元逼近理论，在最小光滑性假设下导出拟最优误差估计。
• 其关键创新在于证明：局部Lipschitz性质足以在不依赖限制性网格角度条件的前提下，获得离散解的L¹控制。
• 该分析结合泛函分析估计与截断技术，避免对逐点离散有界性的依赖。

实验结果

研究问题

• RQ1 能否在不依赖限制性网格角度条件的前提下，为具有临界指数的半线性问题的有限元逼近建立先验误差估计？
• RQ2 是否可能在Galerkin逼近中不强制实施离散最大值原理，而对非线性项进行控制？
• RQ3 非线性项的局部Lipschitz性质是否足以建立拟最优误差估计并实现离散解的L¹控制？
• RQ4 能否仅使用标准有限元逼近理论，在无几何网格约束的前提下获得离散解的L¹有界性？
• RQ5 De Giorgi迭代方法和截断函数在建立临界指数问题的先验L¹有界性中起到何种作用？

主要发现

• 本文在不依赖离散最大值原理的前提下，为Galerkin逼近建立了拟最优先验误差估计。
• 非线性项的局部Lipschitz性质（在临界指数范围内成立）足以控制非线性项，而无需逐点有界性。
• 离散解的L¹控制在不施加限制性角度条件的前提下实现，完全依赖标准有限元逼近理论。
• 该方法适用于2D和3D中具有临界指数的半线性问题，包括几何分析和广义相对论中的重要情形。
• 该方法消除了在非结构化或自适应有限元环境中难以满足的依赖网格角度的条件。
• 结果将有限元方法的适用范围扩展至更广泛的非线性PDE类，在实际计算场景中更具实用性。