Nonoverlapping Domain Decomposition Preconditioners for Discontinuous Galerkin Approximations of Hamilton–Jacobi–Bellman Equations
Nonoverlapping Domain Decomposition Preconditioners for Discontinuous Galerkin Approximations of Hamilton–Jacobi–Bellman Equations
复制标题
Hamilton-Jacobi-Bellman 方程的不连续 Galerkin 近似的非重叠域分解预条件子
DOI：
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发表时间：
2014
影响因子：
2.5
通讯作者：
Iain Smears
中科院分区：
文献类型：
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作者：
Iain Smears
分析了一类由全非线性Hamilton-Jacobi-Bellman （HJB）偏微分方程不连续Galerkin有限元近似引起的非对称线性系统的非重叠区域分解预条件。这些非对称线性系统相对于与矩阵$$mathbf {A}$$ a相关的对称双线性形式是一致有界的和强制的。在这项工作中，我们构造了一个基于$$mathbf {A}$$ a的非重叠域分解预条件$$mathbf {P}$$ P，然后我们证明了该预条件对于解决非对称问题的有效性可以根据条件数$$kappa (mathbf {P}^{-1}mathbf {A})$$ κ(P- 1a)来研究。特别地，我们建立了界$$kappa (mathbf {P}^{-1}mathbf {A})lesssim 1+ p^6 H^3 /q^3 h^3$$ κ(p - 1a)≥1+p6H3/q3h3，其中H和H分别为粗、细网格尺寸，q和p分别为粗、细网格多项式度。这代表了这类方法的第一个结果，它显式地说明了条件数对q的依赖性；我们的分析是建立在原始的精细和粗糙不连续有限元空间之间的最优阶近似结果的基础上的。数值实验证明了该边界的清晰性。虽然预条件对多项式度的鲁棒性不强，但我们的界量化了粗、细空间多项式度的影响。此外，我们通过计算表明，这些方法在实际应用中是有效的，适用于中等多项式次的h-细化下的非对称，全非线性HJB方程。
We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximations 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 $$mathbf {A}$$A. In this work, we construct a nonoverlapping domain decomposition preconditioner $$mathbf {P}$$P, that is based on $$mathbf {A}$$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 $$kappa (mathbf {P}^{-1}mathbf {A})$$κ(P-1A). In particular, we establish the bound $$kappa (mathbf {P}^{-1}mathbf {A})lesssim 1+ p^6 H^3 /q^3 h^3$$κ(P-1A)≲1+p6H3/q3h3, 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.