Dune-UDG: A Cut-Cell Framework for Unfitted Discontinuous Galerkin Methods

Engwer, Christian; Heimann, Felix

doi:10.1007/978-3-642-28589-9_7

Cited by 22 publications

(21 citation statements)

References 14 publications

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“…We also present the UFL scripts that were used to obtain the numerical results illustrating the simplicity by which the end user can access the CutFEM paradigm (Figures 17, 18, and 21). Other software projects including CutFEM style technology include GetFem++ [49], LifeV [50], and DUNE-UDG [51].…”

Section: E Burman Et Almentioning

confidence: 99%

CutFEM: Discretizing geometry and partial differential equations

Burman

Claus

Hansbo

et al. 2014

Numerical Meth Engineering

673

603

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Section: E Burman Et Almentioning

confidence: 99%

CutFEM: Discretizing geometry and partial differential equations

Burman

Claus

Hansbo

et al. 2014

Numerical Meth Engineering

673

603

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“…Such methods were introduced in [3,4] for elliptic equations in curved domains. See also [36,8,31]. In this setting we are concerned with bulk meshes independent of the surface.…”

Section: Introductionmentioning

confidence: 99%

Unfitted Finite Element Methods Using Bulk Meshes for Surface Partial Differential Equations

Deckelnick¹,

Elliott²,

Ranner³

2014

SIAM J. Numer. Anal.

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In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ R n+1 , is embedded in a polyhedral domain in R n+1 consisting of a union, T h , of (n + 1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on T h . Our first method is a sharp interface method, SIF, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γ h , of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [42]. The second method, NBM, is a narrow band method in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [13] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L 2 and H 1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.

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“…This generated code only uses standard C++ components and can easily be used in other projects. The code is in productive use in DUNE for our implementation of the Unfitted Dicontinuous Galerkin method [9].…”

Section: Methodsmentioning

confidence: 99%

Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees

Engwer

Nüßing

2017

ACM Trans. Math. Softw.

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Implicitly described domains are a well established tool in the simulation of time dependent problems, e.g. using level-set methods. In order to solve partial differential equations on such domains, a range of numerical methods was developed, e.g. the Immersed Boundary method, Unfitted Finite Element or Unfitted discontinuous Galerkin methods, eXtended or Generalised Finite Element methods, just to name a few. Many of these methods involve integration over cut-cells or their boundaries, as they are described by sub-domains of the original level-set mesh.We present a new algorithm to geometrically evaluate the integrals over domains described by a first-order, conforming level-set function. The integration is based on a polyhedral reconstruction of the implicit geometry, following the concepts of the Marching Cubes algorithm. The algorithm preserves various topological properties of the implicit geometry in its polyhedral reconstruction, making it suitable for Finite Element computations. Numerical experiments show second order accuracy of the integration.An implementation of the algorithm is available as free software, which allows for an easy incorporation into other projects. The software is in productive use within the DUNE framework [3].

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Dune-UDG: A Cut-Cell Framework for Unfitted Discontinuous Galerkin Methods

Cited by 22 publications

References 14 publications

CutFEM: Discretizing geometry and partial differential equations

CutFEM: Discretizing geometry and partial differential equations

Unfitted Finite Element Methods Using Bulk Meshes for Surface Partial Differential Equations

Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees

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