High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

Fu, Pei; Frachon, Thomas; Kreiss, Gunilla; Zahedi, Sara

doi:10.1007/s10915-021-01756-w

Cited by 5 publications

(1 citation statement)

References 30 publications

(42 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reason is that ghost penalty stabilization [5] has been shown to work well with low as well as high order elements using both continuous and discontinuous elements for bulk problems, see e.g. [6,10,12,14].…”

mentioning

confidence: 99%

A divergence preserving cut finite element method for Darcy flow

Frachon¹,

Hansbo²,

Nilsson³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RT 0 ×P 0 , BDM 1 × P 0 , and RT 1 × P 1 . We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, pollutes the computed velocity field so the divergence-free property of the considered elements is lost. Therefore, we propose two corrections to the standard stabilization strategy; using macro-elements and new stabilization terms for the pressure. By decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is active only where needed. By modifying the standard stabilization terms for the pressure we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. Numerical experiments indicate that with the new stabilization terms the unfitted finite element discretization, for the given element pairs, results in 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative the computational mesh.

show abstract

mentioning

confidence: 99%

A divergence preserving cut finite element method for Darcy flow

Frachon¹,

Hansbo²,

Nilsson³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Accuracy Analysis for Explicit-Implicit Finite Volume Schemes on Cut Cell Meshes

May,

Laakmann

2024

Commun. Appl. Math. Comput.

View full text Add to dashboard Cite

The solution of time-dependent hyperbolic conservation laws on cut cell meshes causes the small cell problem: standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells. In May and Berger (J Sci Comput 71: 919–943, 2017), the mixed explicit-implicit approach in general and MUSCL-Trap (monotonic upwind scheme for conservation laws and trapezoidal scheme) in particular have been introduced to solve this problem by using implicit time stepping on the cut cells. Theoretical and numerical results have indicated that this might lead to a loss in accuracy when switching between the explicit and implicit time stepping. In this contribution, we examine this in more detail and will prove in one dimension that the specific combination MUSCL-Trap of an explicit second-order and an implicit second-order scheme results in a fully second-order mixed scheme. As this result is unlikely to hold in two dimensions, we also introduce two new versions of mixed explicit-implicit schemes based on exchanging the explicit scheme. We present numerical tests in two dimensions where we compare the new versions with the original MUSCL-Trap scheme.

show abstract