### Abstract

In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection. The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.

Original language | English |
---|---|

Pages (from-to) | 191-219 |

Number of pages | 29 |

Journal | Journal of Numerical Mathematics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - Oct 1 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- feedback stabilization
- non-conforming finite elements
- Stokes equations

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

*Journal of Numerical Mathematics*,

*22*(3), 191-219. https://doi.org/10.1515/jnma-2014-0009

**A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations.** / Benner, P.; Saak, J.; Schieweck, F.; Skrzypacz, P.; Weichelt, H. K.

Research output: Contribution to journal › Article

*Journal of Numerical Mathematics*, vol. 22, no. 3, pp. 191-219. https://doi.org/10.1515/jnma-2014-0009

}

TY - JOUR

T1 - A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations

AU - Benner, P.

AU - Saak, J.

AU - Schieweck, F.

AU - Skrzypacz, P.

AU - Weichelt, H. K.

PY - 2014/10/1

Y1 - 2014/10/1

N2 - In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection. The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.

AB - In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection. The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.

KW - feedback stabilization

KW - non-conforming finite elements

KW - Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=84908083896&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908083896&partnerID=8YFLogxK

U2 - 10.1515/jnma-2014-0009

DO - 10.1515/jnma-2014-0009

M3 - Article

VL - 22

SP - 191

EP - 219

JO - Journal of Numerical Mathematics

JF - Journal of Numerical Mathematics

SN - 1570-2820

IS - 3

ER -