Higher order Galerkin–collocation time discretization with Nitsche’s method for the Navier–Stokes equations

Anselmann, Mathias; Bause, Markus

doi:10.1016/j.matcom.2020.10.027

Cited by 13 publications

(22 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, the Dirichlet boundary conditions are enforced in a weak form by adding face integrals to the variational problem. We also refer to [35]. Degrees of freedom assigned to Dirichlet portions of the boundary are then treated as unknowns of the variational problem.…”

Section: Semidiscretization In Space With Weak Enforcement Of Boundar...mentioning

confidence: 99%

“…Futher, discontinuous Galerkin methods offer stronger stability properties since they are known to be strongly A-stable. Applying the discontinuous Galerkin scheme to Problem 3.2 and using a discontinuous local test basis, yields the following time marching scheme; cf., e.g., [35,34].…”

Section: Fully Discrete Problemmentioning

confidence: 99%

“…for a suitably defined nonlinear function R+S) . We refer to [35] for the explicit derivation of the algebraic system of a related space-time finite element approximation of the Navier-Stokes system. To solve the nonlinear problem (3.11), we use Newton's method such that the linear system…”

Section: Fully Discrete Problemmentioning

confidence: 99%

“…where d v n,l and d p n,l , with l ∈ {0, 1}, denote the components of D m n related to velocity and pressure unknowns, respectively. For a more detailed derivation of the the system matrix of (3.15) and the definition of the submatrices B i and F i we refer to [35] again.…”

Section: Fully Discrete Problemmentioning

confidence: 99%

“…Their application puts an additional complexity to the solution of the linear systems due to their more complex block structrue; cf. Table 3.1 and, e.g., [21,35,39]. Here, we use the GMG method as a preconditioner for Krylov subspace iterations, which is a standard concept for the efficient solution of high-dimensional linear systems arising from the discretization of partial differential equations.…”

Section: A Parallel Geometric Multigrid Preconditionermentioning

confidence: 99%

See 4 more Smart Citations

A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

Anselmann¹,

Bause²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier-Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.

show abstract

Section: Semidiscretization In Space With Weak Enforcement Of Boundar...mentioning

confidence: 99%

Section: Fully Discrete Problemmentioning

confidence: 99%

Section: Fully Discrete Problemmentioning

confidence: 99%

Section: Fully Discrete Problemmentioning

confidence: 99%

Section: A Parallel Geometric Multigrid Preconditionermentioning

confidence: 99%

See 3 more Smart Citations

A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

Anselmann¹,

Bause²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Cut finite element methods and ghost stabilization techniques for space‐time discretizations of the Navier–Stokes equations

Anselmann

Bause

2022

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

We propose and analyze numerically a new fictitious domain method, based on higher order space-time finite element discretizations, for the simulation of the nonstationary, incompressible Navier-Stokes equations on evolving domains. The physical domain is embedded into a fixed computational mesh such that arbitrary intersections of the moving domain's boundaries with the background mesh occur. The key ingredients of the approach are the weak formulation of Dirichlet boundary conditions by Nitsche's method, the flexible and efficient integration over all types of intersections of cells by moving boundaries and the spatial extension of the discrete physical quantities to the entire computational background mesh including fictitious (ghost) subdomains of fluid flow. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization ensuring the stability of the approach and defining implicitly the extension to host domains is added. These techniques are embedded in an arbitrary order, discontinuous Galerkin discretization of the time variable and an inf-sup stable discretization of the spatial variables. The convergence and stability properties of the approach are studied, firstly, for a benchmark problem of flow around a stationary obstacle and, secondly, for flow around moving obstacles with arising cut cells and fictitious domains. The parallel implementation is also addressed.

show abstract

A higher order fictitious domain method for the Navier–Stokes equations

Bause

Anselmann

2021

Proc Appl Math and Mech

Self Cite

View full text Add to dashboard Cite

We present a higher order approach for the numerical approximation of the Navier-Stokes equations of incompressible viscous flow on time-dependent domains. A fictitious domain method using non body-fitted finite element meshes for immersed and moving boundaries along with cut cell techniques is applied. Higher order discontinuous Galerkin time discretization schemes and inf-sup stable pairs of finite elements for the discretization in space are further ingredients of the approach.

show abstract

Higher order Galerkin–collocation time discretization with Nitsche’s method for the Navier–Stokes equations

Cited by 13 publications

References 28 publications

A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

Cut finite element methods and ghost stabilization techniques for space‐time discretizations of the Navier–Stokes equations

A higher order fictitious domain method for the Navier–Stokes equations

Contact Info

Product

Resources

About