Convergence analysis of a full discretization of operator-valued differential Riccati equations

TL;DR

This paper proves the convergence of a combined spatial and temporal discretization method for operator-valued differential Riccati equations, demonstrating order one in time and order two in space, which is crucial for practical applications.

Contribution

It provides the first convergence analysis of a full discretization (finite elements and Lie splitting) for DRE, extending previous operator splitting results.

Findings

Convergence with order one in time under weak assumptions.

Convergence with order two in space.

Numerical experiments confirm theoretical results.

Abstract

In recent previous work [E. Hansen, T. Stillfjord and T. \r{A}berg, SIAM J. Numer. Anal., to appear], we analyzed the convergence of operator splitting methods applied to operator-valued differential Riccati equations (DRE). In this paper, we extend these results by analyzing the convergence of a full discretization based on finite elements in space and Lie splitting in time. As far as we are aware, this is the first such analysis for DRE. There are very few analyses of temporal discretizations of DRE overall, and none of them have been combined with spatial discretizations. However, it is clearly vital to know when the full discretization converges, since this is what will be used in practical applications. Our main result is that except for logarithmic factors, the method converges with order one in time and order two in space, under fairly weak assumptions on the problem data. This is illustrated by a numerical experiment based on an application in optimal control.