High order approximations of the operator Lyapunov equation have low rank (Q2100544)

When dealing with non-local operators, low-rank approximation methods are turning out to be a method of choice both for theoretical analysis as well as a foundation for constructing high performance numerical algorithms. This work presents a low-rank greedily adapted \(hp\)-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation. The authors show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high-order finite element spaces. Numerical experiments indicate that they achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in a dumbbell-domain using a finite element space of dimension of only 104 degrees of freedom. Even more surprising is the observation that the \(hp\)-refinement has an effect of reducing the rank of the approximation of the solution.