Volume 36, Issue 6
Finite Element Exterior Calculus for Parabolic Evolution Problems on Riemannian Hypersurfaces

Michael Holst & Christopher Tiee

J. Comp. Math., 36 (2018), pp. 792-832.

Published online: 2018-08

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  • Abstract

Over the last ten years, Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented.


  • Keywords

FEEC Elliptic equations Evolution equations Approximation theory Inf-sup conditions a priori estimates Variational crimes Equations on manifolds

  • AMS Subject Headings

65M15 65M60 53C44

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mholst@math.ucsd.edu (Michael Holst)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-792, author = {Holst , Michael and Tiee , Christopher }, title = {Finite Element Exterior Calculus for Parabolic Evolution Problems on Riemannian Hypersurfaces}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {6}, pages = {792--832}, abstract = {

Over the last ten years, Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented.


}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1705-m2016-0545}, url = {http://global-sci.org/intro/article_detail/jcm/12603.html} }
TY - JOUR T1 - Finite Element Exterior Calculus for Parabolic Evolution Problems on Riemannian Hypersurfaces AU - Holst , Michael AU - Tiee , Christopher JO - Journal of Computational Mathematics VL - 6 SP - 792 EP - 832 PY - 2018 DA - 2018/08 SN - 36 DO - http://dor.org/10.4208/jcm.1705-m2016-0545 UR - https://global-sci.org/intro/jcm/12603.html KW - FEEC KW - Elliptic equations KW - Evolution equations KW - Approximation theory KW - Inf-sup conditions KW - a priori estimates KW - Variational crimes KW - Equations on manifolds AB -

Over the last ten years, Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented.


Michael Holst & Christopher Tiee. (2020). Finite Element Exterior Calculus for Parabolic Evolution Problems on Riemannian Hypersurfaces. Journal of Computational Mathematics. 36 (6). 792-832. doi:10.4208/jcm.1705-m2016-0545
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