Whitney Forms of Higher Degree

Rapetti, Francesca; Bossavit, Alain

doi:10.1137/070705489

Cited by 72 publications

(97 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the mass M and stiffness S matrices appearing on the expression for X should be modified. To derive these matrices, we first recall the expression for the vector proxies of Whitney 1-and 2-forms on triangular meshes [34,35,36]…”

Section: Triangular-element-based Fe Meshesmentioning

confidence: 99%

Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes

Nicolini

Lee

et al. 2020

Journal of Computational Physics

View full text Add to dashboard Cite

Numerical Cherenkov radiation (NCR) or instability is a detrimental effect frequently found in electromagnetic particle-in-cell (EM-PIC) simulations involving relativistic plasma beams. NCR is caused by spurious coupling between electromagnetic-field modes and multiple beam resonances. This coupling may result from the slow down of poorlyresolved waves due to numerical (grid) dispersion and from aliasing mechanisms. NCR has been studied in the past for finite-difference-based EM-PIC algorithms on regular (structured) meshes with rectangular elements. In this work, we extend the analysis of NCR to finite-element-based EM-PIC algorithms implemented on unstructured meshes. The influence of different mesh element shapes and mesh layouts on NCR is studied. Analytic predictions are compared against results from finite-element-based EM-PIC simulations of relativistic plasma beams on various mesh types.

show abstract

Section: Triangular-element-based Fe Meshesmentioning

confidence: 99%

Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes

Nicolini

Lee

et al. 2020

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…Higher order Whitney forms are proposed, for example, in [230,231]. As said, Whitney maps provide a single interpolation within each tetrahedron (if the mesh is three-dimensional).…”

Section: From Cochain To Differential Formmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

View full text Add to dashboard Cite

Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

show abstract

“…We adopt here the high order extension of Nédélec elements presented in Rapetti (2007) and Rapetti and Bossavit (2009). The definition of the basis functions is rather simple since it only involves the barycentric coordinates of the simplex.…”

Section: High Order Edge Finite Elementsmentioning

confidence: 99%