A near-optimal recovery algorithm for the Stokes equations with incomplete information on the boundary conditions

TL;DR

This paper introduces a near-optimal algorithm for approximating velocity and pressure in the Stokes system with incomplete boundary data, leveraging available measurements to produce accurate solutions.

Contribution

The paper presents a novel algorithm that effectively approximates solutions to the Stokes equations using partial boundary information, achieving near-optimal accuracy.

Findings

The algorithm guarantees near-optimal approximation in the energy norm.

It effectively utilizes linear measurements of velocity and pressure.

The approach addresses the challenge of incomplete boundary conditions.

Abstract

We address the problem of numerically approximating the velocity and pressure governed by the Stokes system when the boundary conditions are only partially known and thus do not uniquely determine the velocity-pressure couple. We propose an algorithm that takes advantage of available linear measurements of the velocity and pressure to construct a numerical approximation. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure couple that minimizes, in the energy norm, the distance to all other solutions satisfying the measurements and the Stokes system.