Topics in structure-preserving discretization

Christiansen, Snorre H.; Munthe-Kaas, Hans; Owren, Brynjulf

doi:10.1017/s096249291100002x

Cited by 124 publications

(125 citation statements)

References 132 publications

(153 reference statements)

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“…The tensor product of a differential k-form on some domain and a differential l-form on a second domain may be naturally realized as a differential (k + l)-form on the Cartesian product of the two domains. When this construction is combined with the standard construction of the tensor product of complexes [17,Section 5.7], we are led to a realization of the tensor product of subcomplexes of the de Rham subcomplex on two domains as a subcomplex of the de Rham complex on the Cartesian product of the domains (see also [11,12]). …”

Section: Tensor Products Of Complexes Of Differential Formsmentioning

confidence: 99%

Finite element differential forms on curvilinear cubic meshes and their approximation properties

2014

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We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L 2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence. Mathematics Subject Classification (2010)

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Section: Tensor Products Of Complexes Of Differential Formsmentioning

confidence: 99%

Finite element differential forms on curvilinear cubic meshes and their approximation properties

2014

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“…For the special case of γ ij = 0 our methods coincide with frame-based integrators proposed in [2]. A comprehensive overview is given in [3]. These integrators are defined in terms of differential equations on manifolds, where the righthand side is given Section 18: Numerical methods of differential equations by a solution dependent vector field y (t) = F (y)| y .…”

Section: Connection To Frame-based Integratorsmentioning

confidence: 99%

TVD‐based Finite Volume Methods for Sound‐Advection‐Buyancy Systems

Knoth

Naumann²,

Wensch³

2014

Proc Appl Math and Mech

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Split-explicit Runge-Kutta methods provide an efficient integration procedure for hyperbolic systems with coupled slow and fast wave phenomena. They are generalized to multirate infinitesimal step methods (MIS) in order to develop an order to provide order conditions and to establish stability properties. The construction of MIS methods is based on an underlying Runge-Kutta method. This method is choosen to be total variation diminishing (TVD) to improve the stability properties of the method. Here, the maximum Courant number is improved by a factor of 4.

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“…We consider the mixed formulation of the Hodge-Laplace problem with variable coefficients, described in Arnold et al (2006Arnold et al ( , 2010 and Christiansen et al (2011), which allows one to include the Darcy problem (see Section 2.2.1). Given a non-negative coefficient α ∈ R + and source terms…”

Section: Mixed Formulation Of the Hodge-laplace Problemmentioning

confidence: 99%

“…The spaces P − r Λ k (T h ) are not the only choice. Indeed, in Hiptmair (2002), Arnold et al (2006Arnold et al ( , 2010 and Christiansen et al (2011), the authors present other finite element differential forms to discretize the deterministic Hodge Laplacian.…”

Section: Finite Element Differential Forms and The Discrete Mean Problemmentioning

confidence: 99%

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

Bonizzoni

Buffa²,

Nobile³

2013

IMA Journal of Numerical Analysis

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We study the mixed formulation of the stochastic Hodge-Laplace problem defined on an n-dimensional domain D (n 1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (m 1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.Keywords: finite element exterior calculus; Hodge Laplacian; mixed finite elements; uncertainty quantification; stochastic partial differential equations; moment equations; sparse tensor product approximation.

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Topics in structure-preserving discretization

Cited by 124 publications

References 132 publications

Finite element differential forms on curvilinear cubic meshes and their approximation properties

Finite element differential forms on curvilinear cubic meshes and their approximation properties

TVD‐based Finite Volume Methods for Sound‐Advection‐Buyancy Systems

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

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