The Time-Domain Cell Method Is a Coupling of Two Explicit Discontinuous Galerkin Schemes With Continuous Fluxes

Kapidani, Bernard; Codecasa, Lorenzo; Specogna, Ruben

doi:10.1109/tmag.2019.2952015

Cited by 8 publications

(6 citation statements)

References 9 publications

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“…is a one-form which coincides exactly with the basis-functions introduced in [10] directly in the global coordinates. As anticipated in the introduction, a recent equivalence proof (in [23]) between the lowest order 3D CM and a DG approach was a leading cause for the present developments.…”

Section: The Lowest Order Element and The Cell Methodsmentioning

confidence: 66%

“…three irregular quadrilaterals, which will be of utmost importance, and we will call fundamental 2-cells (see the definition of micro-cell in [26] or see [23]) and denote with K in the rest of the paper. Definitions of lower dimensional sets K Ω 1 and K Ω 0 are intuitive, but we additionally provide here an explicit decomposition of K Ω 1 into the two sets of segments…”

Section: Barycentric-dual Complexesmentioning

confidence: 99%

“…In the present paper we are strongly inspired by this latter framework: we start from the set of basis-functions introduced by Codecasa and co-authors in [10,11], and more recently studied in [23], where an equivalence between their formulation and a peculiar DG one using two barycentric-dual unstructured meshes and piecewise-constant basis-functions was proven by some of the present authors. Building on this result, we show how to extend the method to arbitrary degree in the local polynomial basis-functions.…”

Section: Introductionmentioning

confidence: 99%

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An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

Kapidani,

Codecasa,

Schöberl

2020

Preprint

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We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis-functions yields cheaply invertible blockdiagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.

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Section: The Lowest Order Element and The Cell Methodsmentioning

confidence: 66%

Section: Barycentric-dual Complexesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

Kapidani,

Codecasa,

Schöberl

2020

Preprint

Self Cite

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show abstract

“…We have presented a DG method for the time dependent ME combining an orthogonal modal basis with the appropriate transformation rules, under changes of coordinates, for the physical fields involved. This combination is fruitful since it yields a very low storage implementation (competitive with the one produced for the nodal basis in [29] or for the Cell Method in [30], [31]) which additionally has only moderate increase in the number of unknowns when a PML is used to truncate an unbounded computational domain. The optimization of our PML and its efficient generalization to arbitrary damping profile and curved elements is a current object of study and will be reported elsewhere.…”

Section: Discussionmentioning

confidence: 99%

A matrix-free Discontinuous Galerkin method for the time dependent Maxwell equations in unbounded domains

Kapidani¹,

Schöberl²

2020

Preprint

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A Discontinuous Galerkin (DG) Finite Element Method (FEM) approach for the 3D time dependent Maxwell equations in unbounded domains is presented. The method (implemented in the FEM library NGsolve) is based on the covariant transformation of a modal orthogonal polynomial basis, originally defined on a reference simplex. The approach leads to an explicit time stepping scheme for which the mass matrix to be inverted is at most d × d block-diagonal (in d = 2, 3 spatial dimensions) while the matrix which discretizes the curl operators on the right-hand side of the system is a small reference matrix, independent from geometric properties of mesh elements. Furthermore, we show that the introduced optimizations are preserved when unbounded domains are also included in the formulation through a complex-stretching based approach.

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“…In addition to being currently limited to two-dimensional manifolds, the construction does not lend itself to higher orders of approximation. One of the present authors has recently introduced an extension of the cell method for the two dimensional Maxwell initial value problem, achieving arbitrary order of convergence and block diagonal mass matrices [35], but since the approach is based on non-conforming spaces [36], sequence properties are not trivially preserved, and differential operators are not clearly related to the topology of an underlying mesh. In any case, the explicit geometric construction of a topological dual mesh can be a cumbersome procedure which adds one layer of complexity to the algorithm and usually introduces more unknowns.…”

mentioning

confidence: 99%

High order geometric methods with splines: an analysis of discrete Hodge--star operators

Kapidani¹,

Vázquez²

2022

Preprint

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A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence tX k h u n k"0 , starting from splines of degree p, and a dual sequence t r X k h u n k"0 , starting from splines of degree p ´1. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge-star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.

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The Time-Domain Cell Method Is a Coupling of Two Explicit Discontinuous Galerkin Schemes With Continuous Fluxes

Cited by 8 publications

References 9 publications

An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

A matrix-free Discontinuous Galerkin method for the time dependent Maxwell equations in unbounded domains

High order geometric methods with splines: an analysis of discrete Hodge--star operators

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