# Generalized parametric solutions in Stokes flow

## TL;DR

This paper develops a PGD-based method for efficiently solving parametric Stokes flow problems, enabling rapid evaluations for design optimization and uncertainty quantification in computational mechanics.

## Contribution

It introduces a PGD formulation tailored for the saddle point structure of the Stokes problem, analyzing different separated forms and demonstrating effectiveness through numerical examples.

## Key findings

- Successful application to Stokes and Brinkman models
- Efficient parametric solutions with reduced computational cost
- Enhanced capability for design optimization and uncertainty quantification

## Abstract

Design optimization and uncertainty quantification, among other applications of industrial interest, require fast or multiple queries of some parametric model. The Proper Generalized Decomposition (PGD) provides a separable solution, a \emph{computational vademecum} explicitly dependent on the parameters, efficiently computed with a greedy algorithm combined with an alternated directions scheme and compactly stored. This strategy has been successfully employed in many problems in computational mechanics. The application to problems with saddle point structure raises some difficulties requiring further attention. This article proposes a PGD formulation of the Stokes problem. Various possibilities of the separated forms of the PGD solutions are discussed and analyzed, selecting the more viable option. The efficacy of the proposed methodology is demonstrated in numerical examples for both Stokes and Brinkman models.