Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

Holst, Michael; Stern, Ari

doi:10.1007/s10208-011-9110-8

Cited by 9 publications

(16 citation statements)

References 28 publications

(52 reference statements)

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“…An explanation of how these estimates are derived from the results of [21] is given in Appendix A. Note that these estimates are exactly the same as the corresponding estimates (4.11), (4.14) and (4.18) from the linear case.…”

Section: Semi-linear Evolution Problemsmentioning

confidence: 61%

“…We have also seen in this article how the recent generalizations of the FEEC by Holst and Stern [20,21] for semi-linear elliptic PDE can be extended to evolution PDE as well, both parabolic and hyperbolic types. We also anticipate that the basic approach to analyzing variational crimes in [20,21] for the linear and semilinar elliptic cases will also work in the case of evolution problems; we will explore the question of variational crimes in a subsequent article, with the target being the analysis of surface finite element methods for evolution problems.…”

Section: Discussionmentioning

confidence: 81%

“…The proof is very similar to that of Theorem 4.2. Equation (6.4) is the k = n case of the discrete mixed variational problem examined by Holst and Stern in [21,Equation (9)]. Therefore, we can use the same type of triangle inequality from (4.10) to recover the estimates.…”

Section: Semi-linear Evolution Problemsmentioning

confidence: 99%

“…Our results recover the estimates of Thomée and Geveci for two-dimensional domains and the lowest-order mixed method as a special case, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. In Section 6, we employ the results of Holst and Stern [21] to extend our parabolic estimates to a class of semi-linear evolution PDE, as encapsulated by Theorem 6.2. Finally, in Section 7, we draw conclusions and make remarks on future directions.…”

Section: Introductionmentioning

confidence: 99%

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Finite Element Exterior Calculus for Evolution Problems

Gillette¹,

Holst

Zhu

2017

JCM

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Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 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, Holst 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 Thomée 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 Thomée 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 Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

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Section: Semi-linear Evolution Problemsmentioning

confidence: 61%

Section: Discussionmentioning

confidence: 81%

Section: Semi-linear Evolution Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finite Element Exterior Calculus for Evolution Problems

Gillette¹,

Holst

Zhu

2017

JCM

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“…The method is formulated on a "flat" triangulated approximation of the curved domain surface, and the error produced by this approximation is then controlled using a "variational crimes" framework known as the Strang Lemmas. Our recent work in this area leverages the Finite Element Exterior Calculus framework (FEEC) [46] to provide a more general error analysis framework for surface finite element methods on n-surfaces, for static linear and nonlinear problems [47,48], as well as for evolution problems on surfaces [49,50]. Surface finite element methods for geometric PDE have the advantage of allowing for the use of standard finite element software originally developed for standard (non-geometric) PDE problems in two-dimensional "flat" domains or three-dimensional volumes, after a fairly simple modification to the reference element maps commonly used by such software packages.…”

Section: Coupled Volume and Surface Diffusion Modelmentioning

confidence: 99%

GAMer 2: A System for 3D Mesh Processing of Cellular Electron Micrographs

Lee

Laughlin

Beaumelle

et al. 2019

Preprint

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Recent advances in electron microscopy have, for the first time, enabled imaging of single cells in 3D at a nanometer length scale resolution. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations will require a system for going from electron micrographs to 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method (FEM). In this paper, we develop an end-to-end pipeline for this task by adapting and extending computer graphics mesh processing and smoothing algorithms. Our workflow makes use of our recently rewritten mesh processing software, GAMer 2, which implements several mesh conditioning algorithms and serves as a platform to connect different pipeline steps. We apply this pipeline to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and demonstrate that the resultant meshes are suitable for finite element simulations. Our pipeline, which consists of free and open-source community driven tools, is a step towards routine physical simulations of biological processes in realistic geometries. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery. Author summary3D imaging of cellular components and associated reconstruction methods have made great strides in the past decade, opening windows into the complex intraceullar organization. These advances also mean that computational tools need to be developed to work with these images not just for purposes of visualization but also for biophysical simulations. Here, we describe a pipeline that takes images from electron microscopy as input and produces smooth surface and volume meshes as output. These meshes are suitable for building high-quality finite element simulations of cellular processes modeled by ordinary and partial differential equations, bringing us closer to realizing the goal of generating high-resolution simulations of such phenomena in realistic geometries. We demonstrate the utility of this pipeline by meshing 3D reconstructions of dendritic spines, calculating 1 the curvatures of the different component membranes, and conducting finite-element simulations of reaction-diffusion equations using the generated meshes. The software tools employed in our pipeline are community driven, open source, and free. We believe that technologies such as those presented will enable a new frontier in biophysical simulations in realistic geometries.

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Constructing reference metrics on multicube representations of arbitrary manifolds

Lindblom

Taylor

Rinne

2016

Journal of Computational Physics

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Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).

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Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

Cited by 9 publications

References 28 publications

Finite Element Exterior Calculus for Evolution Problems

Finite Element Exterior Calculus for Evolution Problems

GAMer 2: A System for 3D Mesh Processing of Cellular Electron Micrographs

Constructing reference metrics on multicube representations of arbitrary manifolds

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