We present a new high-order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as zero level set function. The is based trace technique. higher accuracy obtained by using an isoparametric mapping volume mesh, function, introduced in [C. Lehrenfeld, Comp. Meth. Appl. Mech. Engrg., 300 (2016), pp. 716--733]. resulting easy to implement. error analysis this and derive optimal order $H^1(\Gamma)$-norm...

The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. approach uses completely Eulerian description domain motion. physical is embedded in triangulated computational and can overlap time-independent background mesh an arbitrary way. based difference discretizations time derivatives standard geometrically unfitted with additional stabilization term spatial domain. performance analysis rely fundamental extension...

In the context of unfitted finite element discretizations, realization high-order methods is challenging due to fact that geometry approximation has be sufficiently accurate. We consider a new method achieves for domains are implicitly described by smooth-level set functions. The based on parametric mapping, which transforms piecewise planar interface reconstruction approximation. Both components, and easy implement. this article, we present an priori error analysis applied problem. reveals...

The vertically upward Taylor flow in a small square channel (side length 2 mm) is one of the guiding measures within priority program “Transport Processes at Fluidic Interfaces” (SPP 1506) German Research Foundation (DFG). This paper presents results coordinated experiments and three-dimensional numerical simulations (with three different academic computer codes) for typical local parameters (bubble shape, thickness liquid film, velocity profiles) cutting planes (lateral diagonal) specific...

We consider a standard model for mass transport across an evolving interface. The solution has to satisfy jump condition present and analyze finite element discretization method this problem. This is based on space-time approach in which discontinuous Galerkin (DG) technique combined with extended (XFEM). satisfied weak sense by using the Nitsche method. XFEM-DG new. An error analysis presented. Results of numerical experiments are given illustrate accuracy

We propose a new discretization method for the Stokes equations. The is an improved version of recently presented in [C. Lehrenfeld and J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which based on $H({div})$-conforming finite element space hybrid discontinuous Galerkin (HDG) formulation viscous forces. $H({div})$-conformity results favorable properties such as pointwise divergence-free solutions pressure robustness. However, approximation velocity with polynomial degree $k$, it...

In this paper, we study a new numerical method for the solution of partial differential equations on evolving surfaces.The is built stabilized trace finite element (TraceFEM) spatial discretization and differences time discretization.The TraceFEM uses stationary background mesh, which can be chosen independent position surface.The stabilization ensures wellconditioning algebraic systems defines regular extension from surface to its volumetric neighborhood.Having such an essential...

Abstract We analyse a Eulerian finite element method, combining time-stepping scheme applied to the time-dependent Stokes equations with CutFEM approach using inf-sup stable Taylor–Hood elements for spatial discretization. This is based on method introduced by Lehrenfeld & Olshanskii (2019, A PDEs in domains. ESAIM: M2AN, 53, 585–614) context of scalar convection–diffusion problems moving domains, and extended nonstationary problem domains Burman et al. arXiv:1910.03054 [math.NA])...

ngsxfem is an add-on library to Netgen/NGSolve, a general purpose, high performance finite element for the numerical solution of partial differential equations.The enables use geometrically unfitted technologies known under different labels, e.g.XFEM, CutFEM, TraceFEM, Finite Cell, fictitious domain method or Cut-Cell methods, etc.. Both, Netgen/NGSolve and are written in C++ with rich Python interface through which it typically used.ngsxfem academic software.Its primary intention facilitate...

Abstract In Heimann, Lehrenfeld, and Preuß (2023, SIAM J. Sci. Comp., 45(2), B139–B165), new geometrically unfitted space–time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space time have been introduced. For a parametric mapping background tensor-product mesh has used. this paper, we concentrate the geometrical approximation derive rigorous bounds distance between realized an ideal different norms results regularity mapping....

In a recent paper [C. Lehrenfeld and A. Reusken, SIAM J. Numer. Anal., 51 (2013), pp. 958--983] new finite element discretization method for class of two-phase mass transport problems is presented analyzed. The problem describes in domain with an evolving interface. Across the interface jump condition has to be satisfied. that space-time approach which combines discontinuous Galerkin (DG) technique (in time) extended (XFEM). Using Nitsche enforced weak sense. While emphasis was on analysis...

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust the under-resolved regime, mass conservation as well energy stability are key ingredients obtain and discretisations. Recently, two approaches have been proposed context high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On one hand, standard $L^2$-based DG enforce weakly by use...

Abstract In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using shape functions that are chosen to be elementwise in the kernel of corresponding operator. We propose new variant, embedded method, which projection an underlying method onto subspace Trefftz‐type. The can described very general way and obtain it no have calculated explicitly, instead embedding operator constructed. simplest cases recovers established methods. But approach allows...

Abstract We introduce a new discretization based on polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this fulfill equations pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, strong reduction degrees freedom is achieved, especially higher degrees. In addition, in contrast many other our approach allows us easily incorporate inhomogeneous right-hand sides (driving forces) by using concept...

In this paper, we propose new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem domain is studied. For accuracy, apply parametric mapping background tensor-product mesh. Concerning discretisation time, consider discontinuous Galerkin, as well related continuous (Petrov-)Galerkin Galerkin collocation methods. stabilisation with...

Summary In this work we consider the numerical solution of incompressible flows on two‐dimensional manifolds. Whereas compatibility demands velocity and pressure spaces are known from flat case one further has to deal with approximation a field that lies only in tangential space given geometry. Abandoning H 1 ‐conformity allows us construct finite elements which are—due an application Piola transformation—exactly tangential. To reintroduce continuity (in weak sense) make use (hybrid)...