The Geometric Basis of Numerical Methods

Gerritsma, Marc; Hiemstra, René R.; Kreeft, Jasper; Palha, Artur; Rebelo, Pedro Pinto; Toshniwal, Deepesh

doi:10.1007/978-3-319-01601-6_2

Cited by 14 publications

(13 citation statements)

References 31 publications

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“…Possible extensions to spectral methods were described by Robidoux, [109]. A different approach for constructing arbitrary order mimetic finite elements has been proposed by the authors [31,64,92,96], with applications to advection problems [95], Stokes' flow [81], MHD equilibrium [94], Navier-Stokes [93], and within a Least-Squares finite element formulation [16,62,63,91].…”

Section: Overview Of Mimetic Discretizationsmentioning

confidence: 99%

Mimetic Spectral Element Method for Anisotropic Diffusion

Gerritsma

Palha

Jain

et al. 2018

Numerical Methods for PDEs

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This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids.

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Section: Overview Of Mimetic Discretizationsmentioning

confidence: 99%

Mimetic Spectral Element Method for Anisotropic Diffusion

Gerritsma

Palha

Jain

et al. 2018

Numerical Methods for PDEs

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“…This is favourable in terms of commutative properties of the operators. Namely, the exterior derivative commutes with pullbacks by smooth functions while the co-derivative only commutes with pullbacks by isometries (Gerritsma et al, 2014). Commutation under pullbacks is an essential testing ground for a discretization:…”

Section: Weak Formulationmentioning

confidence: 99%

An electrical engineering perspective on missed opportunities in computational physics

Lahtinen,

Kotiuga,

Stenvall

2018

Preprint

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We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We emphasize the virtues of the concept of elliptic complex and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry and modelling software design. In particular, the ubiquitous concept of naturality is central. We discuss the Galerkin finite element method as a way to achieve a discrete formulation and discuss its compatibility with so-called cochain methods. Despite the apparent differences in their underlying principles, in both one finds a finite-dimensional subcomplex of a cochain complex. From such a viewpoint, compatibility of a discretization boils down to preserving properties in such a process. Via reflection on the historical background and the identification of common structures, forward-looking research questions may be framed.

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“…In this section a brief introduction to the discretization of physical quantities and to the discretization of the exterior derivative is presented. For a more detailed presentation the reader is directed to [2,4,6] Consider a three dimensional domain Ω and an associated grid consisting of a collection of points, τ (0),i , line segments connecting the points, τ (1),i , surfaces bounded by these line segments, τ (2),i , and volumes bounded by these surfaces, τ (3),i .…”

Section: Mimetic Discretizationmentioning

confidence: 99%

“…is the incidence matrix containing only the values, 0, 1 and -1, see [2,4,6] for more details. This idea can be extended to higher dimensions, giving rise to k-incidence matrices, E (k+1,k) i j , which represent the discrete exterior derivative on discrete k-forms, see [2,6].…”

Section: Mimetic Discretizationmentioning

confidence: 99%

Mimetic Spectral Element Advection

Palha

Rebelo

Gerritsma

2013

Lecture Notes in Computational Science and Engineering

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We present a discretization of the linear advection of differential forms on bounded domains. The framework established in [4] is extended to incorporate the Lie derivative, L, by means of Cartan's homotopy formula. The method is based on a physics-compatible discretization with spectral accuracy . It will be shown that the derived scheme has spectral convergence with local mass conservation. Artificial dispersion depends on the order of time integration.

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The Geometric Basis of Numerical Methods

Cited by 14 publications

References 31 publications

Mimetic Spectral Element Method for Anisotropic Diffusion

Mimetic Spectral Element Method for Anisotropic Diffusion

An electrical engineering perspective on missed opportunities in computational physics

Mimetic Spectral Element Advection

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