ngsxfem: Add-on to NGSolve for geometrically unfitted finite element discretizations

Lehrenfeld, Christoph; Heimann, Fabian; Preuß, Janosch; Wahl, Henry von

doi:10.21105/joss.03237

Cited by 23 publications

(16 citation statements)

References 7 publications

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“…In the implementation of the discretizations (P1h) and (P2h) of the surface Stokes problem, we use (unless stated otherwise) the parameter setting and components listed in a)-d) at the end of section 6 (with a constant 1 in ∼). The methods are implemented in Netgen/NGSolve with ngsxfem [1,18].…”

Section: The Discrete Problem Reads: Determinementioning

confidence: 99%

Higher order Trace Finite Element Methods for the Surface Stokes Equation

Jankuhn,

Reusken

2019

Preprint

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In this paper a class of higher order finite element methods for the discretization of surface Stokes equations is studied. These methods are based on an unfitted finite element approach in which standard Taylor-Hood spaces on an underlying bulk mesh are used. For treating the constraint that the velocity must be tangential to the surface a penalty method is applied. Higher order geometry approximation is obtained by using a parametric trace finite element technique, known from the literature on trace finite element methods for scalar surface partial differential equations. Based on theoretical analyses for related problems, specific choices for the parameters in the method are proposed. Results of a systematic numerical study are included in which different variants are compared and convergence properties are illustrated.

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Section: The Discrete Problem Reads: Determinementioning

confidence: 99%

Higher order Trace Finite Element Methods for the Surface Stokes Equation

Jankuhn,

Reusken

2019

Preprint

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“…In this section we present results of a few numerical experiments. We implemented the different methods in Netgen/NGSolve with ngsxfem [1,13].…”

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confidence: 99%

Trace Finite Element Methods for Surface Vector-Laplace Equations

Jankuhn¹,

Reusken²

2019

Preprint

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In this paper we analyze a class of trace finite element methods (TraceFEM) for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface ("tangent condition"). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. A main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of Lagrange multiplier.

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“…We consider three problem geometries to investigate specific aspects, each in one of the following subsections. Recall that the notation k = r means k s = k t = r and analogously q = r means q s = q t = r. All experiments are performed with ngsxfem [21], an unfitted finite element extension of ngsolve. Reproduction data are available at https://gitlab.gwdg.de/fabian.…”

Section: Numerical Investigationsmentioning

confidence: 99%

Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

Heimann¹,

Lehrenfeld²,

Preuß³

2022

Preprint

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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 on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretisation in time, we consider discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and Galerkin collocation methods. For stabilisation with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilisation is employed. The article puts an emphasis on the techniques that allow to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

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ngsxfem: Add-on to NGSolve for geometrically unfitted finite element discretizations

Cited by 23 publications

References 7 publications

Higher order Trace Finite Element Methods for the Surface Stokes Equation

Higher order Trace Finite Element Methods for the Surface Stokes Equation

Trace Finite Element Methods for Surface Vector-Laplace Equations

Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

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