用于 Hamilton–Jacobi–Bellman 方程间断 Galerkin 逼近的非重叠区域分解预条件子
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations
AI总结：
本文针对 HJB 方程间断 Galerkin 离散产生的非对称系统，构造基于对称矩阵 A 的非重叠区域分解预条件子，证明含粗细网格尺寸与多项式次数的条件数界，并用数值实验验证其尖锐性和实用性。
摘要：
我们分析了一类非重叠区域分解预条件子，它们用于由完全非线性 Hamilton–Jacobi–Bellman（HJB）偏微分方程的间断 Galerkin 有限元逼近所产生的非对称线性系统。这些非对称线性系统相对于与矩阵 A 相关联的一个对称双线性形式是一致有界且强制的。在本文中，我们构造了一个基于 A 的非重叠区域分解预条件子 P，随后表明，该预条件子求解非对称问题的有效性可以通过条件数 κ(P^{-1}A) 来研究。特别地，我们建立了界 κ(P^{-1}A) ≲ 1+p^6H^3/q^3h^3，其中 H 和 h 分别为粗网格和细网格尺寸，q 和 p 分别为粗网格和细网格的多项式次数。这是此类方法首个显式说明条件数对 q 依赖性的结果；我们的分析建立在细、粗间断有限元空间之间一个原创的最优阶逼近结果之上。数值实验证明了该界的尖锐性。尽管这些预条件子关于多项式次数并不稳健，但我们的界量化了粗空间和细空间多项式次数的影响。此外，我们通过计算表明，在中等多项式次数下进行 h 加密时，这些方法在非对称、完全非线性 HJB 方程的实际应用中是有效的。
英文摘要：
We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix $\mathbf{A}$. In this work, we construct a nonoverlapping domain decomposition preconditioner $\mathbf{P}$, that is based on $\mathbf{A}$, and we then show that the effectiveness of the preconditioner for solving the} nonsymmetric problems can be studied in terms of the condition number $κ(\mathbf{P}^{-1}\mathbf{A})$. In particular, we establish the bound $κ(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first such result for this class of methods that explicitly accounts for the dependence of the condition number on $q$; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under $h$-refinement for moderate polynomial degrees.
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