Robust discretization and solvers for elliptic optimal control problems with energy regularization

TL;DR

This paper develops robust finite element discretization and iterative solvers for singularly perturbed elliptic equations from optimal control problems with energy regularization, ensuring optimal convergence and efficient parallel solution.

Contribution

It introduces quasi-optimal error estimates depending on mesh size and regularization, and analyzes the robustness and parallel performance of algebraic multigrid and BDDC preconditioners.

Findings

Optimal convergence achieved with regularization parameter proportional to mesh size squared

Preconditioners demonstrate robustness across mesh sizes and regularization parameters

Parallel BDDC solver shows efficient scalability and performance

Abstract

We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for elliptic distributed optimal control problems with energy regularization that were recently studied by M.~Neum\"{u}ller and O.~Steinbach (2020). We provide quasi-optimal a priori finite element error estimates which depend both on the mesh size $h$ and on the regularization parameter $\varrho$. The choice $\varrho = h^2$ ensures optimal convergence which only depends on the regularity of the target function. For the iterative solution, we employ an algebraic multigrid preconditioner and a balancing domain decomposition by constraints (BDDC) preconditioner. We numerically study robustness and efficiency of the proposed algebraic preconditioners with respect to the mesh size $h$, the regularization parameter $\varrho$, and the number of subdomains (cores) $p$. Furthermore, we investigate the parallel performance of the BDDC preconditioned conjugate gradient solver.