Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation

TL;DR

This paper develops a reduced order modeling approach for the parametrized Stokes equation using a discontinuous Galerkin method, enabling efficient simulations on geometrically varying domains.

Contribution

It introduces a novel combination of discontinuous Galerkin discretization with proper orthogonal decomposition for geometric parametrization in Stokes flow.

Findings

Effective error reduction demonstrated

Eigenvalue decay indicates compact basis

Significant online computational speedup

Abstract

The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time. Keywords: Discontinuous Galerkin method, Stokes flow, Geometric parametrization, Proper orthogonal decomposition