Convergence analysis of a balancing domain decomposition method for an elliptic optimal control problem with HDG discretizations

TL;DR

This paper analyzes the convergence of a BDDC domain decomposition method applied to an elliptic optimal control problem discretized with HDG, showing robustness and efficiency through theoretical and numerical results.

Contribution

It provides a convergence analysis of BDDC for HDG discretizations in elliptic control problems, demonstrating robustness and iteration independence from the number of subdomains.

Findings

Algorithm is robust with respect to the regularization parameter.

Number of iterations is independent of the number of subdomains.

Numerical experiments confirm theoretical convergence results.

Abstract

In this work, a balancing domain decomposition by constraints (BDDC) algorithm is applied to the nonsymmetric positive definite linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of an elliptic distributed optimal control problem. Convergence analysis for the BDDC preconditioned generalized minimal residual (GMRES) solver demonstrates that, when the subdomain size is small enough, the algorithm is robust with respect to the regularization parameter, and the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. Numerical experiments are performed to confirm the theoretical results.