Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

TL;DR

The paper addresses the solution of nonsymmetric linear systems from high-order DG discretizations of fully nonlinear HJB equations by developing nonoverlapping additive Schwarz preconditioners based on the symmetric form $a_h$. A central contribution is a spectral bound on the preconditioned system, $\kappa(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ \frac{p^2 H}{q h} + \frac{p^6 H^3}{q^3 h^3}$, which explicitly reveals the influence of coarse-space degree $q$ alongside the mesh sizes $H$, $h$ and polynomial degrees $p$. This bound rests on an original optimal-order approximation result between the fine space $V_{h,\mathbf{p}}$ and the coarse space $V_{H,\mathbf{q}}$, enabling a stable decomposition essential to Schwarz theory. Numerical experiments verify the sharpness of the bound and demonstrate the method’s practicality for $h$-refinement with moderate $p$, including comparisons with overlapping methods and applications to HJB problems, where the preconditioners remain effective despite anisotropy and nonlinearity. Overall, the work provides a concrete, provable framework for efficiently preconditioning DG discretizations of HJB equations, with clear guidance on how coarse-space polynomial degree impacts performance in practice.

Abstract

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.

Figures (1)

• Figure 1: Overlapping and nonoverlapping decompositions of $\Omega=(0,1)^2$ used in the experiment of section \ref{['sec:numexp2']}. Four subdomains are used for both the overlapping and nonoverlapping methods, with the overlap size $\delta$ defined as the length shown above.

Theorems & Definitions (21)

• lemma 1
• proof
• theorem 1
• Remark 1
• theorem 2
• Remark 2
• theorem 3
• lemma 2
• lemma 3
• proof