### Abstract

This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of time-dependent partial differential equations. Using the heat equation and the time-dependent Stokes’ equation as examples, we discuss how accurately a FOSLS finite-element formulation adheres to the underlying energy law associated with the physical system. Using regularity arguments involving the initial condition of the system, we are able to give bounds on the convergence of the discrete energy law to its expected value (zero in the examples presented here). Numerical experiments are performed, showing that the discrete energy laws hold with order O(h^{2} ^{p}), where h is the mesh spacing and p is the order of the finite-element space. Thus, the energy law conformance is held with a higher order than the expected, O(h^{p}), convergence of the finite-element approximation. Finally, we introduce an abstract framework for analyzing the energy laws of general FOSLS discretizations.

Original language | English (US) |
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Title of host publication | Large-Scale Scientific Computing - 11th International Conference, LSSC 2017, Revised Selected Papers |

Editors | Ivan Lirkov, Svetozar Margenov |

Publisher | Springer Verlag |

Pages | 3-20 |

Number of pages | 18 |

ISBN (Print) | 9783319734408 |

DOIs | |

Publication status | Published - Jan 1 2018 |

Event | 11th International Conference on Large-Scale Scientific Computations, LSSC 2017 - Sozopol, Bulgaria Duration: Sep 11 2017 → Sep 15 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10665 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 11th International Conference on Large-Scale Scientific Computations, LSSC 2017 |
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Country | Bulgaria |

City | Sozopol |

Period | 9/11/17 → 9/15/17 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Large-Scale Scientific Computing - 11th International Conference, LSSC 2017, Revised Selected Papers*(pp. 3-20). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10665 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-73441-5_1