### Abstract

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Hoist and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomee for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Hoist and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

Original language | English (US) |
---|---|

Pages (from-to) | 187-212 |

Number of pages | 26 |

Journal | Journal of Computational Mathematics |

Volume | 35 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2017 |

### Fingerprint

### Keywords

- A priori estimates
- Approximation theory
- Elliptic equations
- Evolution equations
- FEEC
- Inf-sup conditions
- Nonlinear approximation
- Nonlinear equations

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

*Journal of Computational Mathematics*,

*35*(2), 187-212. https://doi.org/10.4208/jcm.1610-m2015-0319

**Finite element exterior calculus for evolution problems.** / Gillette, Andrew; Hoist, Michael; Zhu, Yunrong.

Research output: Contribution to journal › Article

*Journal of Computational Mathematics*, vol. 35, no. 2, pp. 187-212. https://doi.org/10.4208/jcm.1610-m2015-0319

}

TY - JOUR

T1 - Finite element exterior calculus for evolution problems

AU - Gillette, Andrew

AU - Hoist, Michael

AU - Zhu, Yunrong

PY - 2017/3/1

Y1 - 2017/3/1

N2 - Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Hoist and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomee for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Hoist and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

AB - Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Hoist and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomee for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Hoist and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

KW - A priori estimates

KW - Approximation theory

KW - Elliptic equations

KW - Evolution equations

KW - FEEC

KW - Inf-sup conditions

KW - Nonlinear approximation

KW - Nonlinear equations

UR - http://www.scopus.com/inward/record.url?scp=85017294654&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017294654&partnerID=8YFLogxK

U2 - 10.4208/jcm.1610-m2015-0319

DO - 10.4208/jcm.1610-m2015-0319

M3 - Article

VL - 35

SP - 187

EP - 212

JO - Journal of Computational Mathematics

JF - Journal of Computational Mathematics

SN - 0254-9409

IS - 2

ER -