A Non-standard Finite Element Method Based on Boundary Integral Operators

Hofreither, Clemens; Langer, Ulrich; Pechstein, Clemens

doi:10.1007/978-3-642-29843-1_3

Cited by 18 publications

(43 citation statements)

References 13 publications

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“…The numerical results have not only confirmed this enhanced stability property of the discretization scheme, but have also indicated faster convergence of the GMRES solver in comparison with the original BEM-based FEM scheme presented in [16,14].…”

Section: Resultsmentioning

confidence: 55%

“…There is a close relation between this BEM-based FEM with piecewise linear boundary data and the so-called method of residual-free bubbles [2,3,5,10,4]. Indeed, it has been shown in [14] that the BEM-based FEM, with exact evaluation of the Steklov-Poincaré operator, is equivalent to the method of residual-free bubbles with exactly computed bubbles. Since the latter has been shown to be a stable method for convection-dominated problems, it seems clear that also the BEM-based FEM should have advantageous stability properties.…”

Section: Introductionmentioning

confidence: 90%

“…Another preprocessing step is the computation of the matrices arising from the local boundary integral formulations. Here, we use the BEM code developed in the PhD thesis by C. Hofreither [14], which is based on a fully numerical integration scheme described in [28]. The inversions of the local single layer potential matrices V T are performed with an efficient LAPACK routine.…”

Section: Preprocessingmentioning

confidence: 99%

“…A BEM-based FEM can also be considered a local Trefftz FEM. The papers [15] and [13] provide the a priori discretization error analysis with respect to the energy and L 2 norms, respectively, where homogeneous diffusion problems serve as model problems. In [25], a new construction of the approximation space was proposed, which employs the polygonal faces of the polyhedral elements.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Convection-adapted BEM-based FEM

Hofreither

Langer

Weißer

2016

Z. Angew. Math. Mech.

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We present a new discretization method for homogeneous convectiondiffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.

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Section: Resultsmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 90%

Section: Preprocessingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convection-adapted BEM-based FEM

Hofreither

Langer

Weißer

2016

Z. Angew. Math. Mech.

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“…The other possible approach is to employ basis functions that satisfy the differential equation locally [29,30]. This method has been studied in detail in [31,32] and extended to higher order polygons in [5,33]. Zienkiewicz [34] presented a concise discussion on different approximation procedures to differential equations.…”

mentioning

confidence: 99%

Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

Hirshikesh

Natarajan

Annabattula

et al. 2017

Asia Pac. J. Comput. Engin.

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Higher order Trefftz‐like Finite Element Method on meshes with L‐shaped elements

Weißer

2014

Proc Appl Math and Mech

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In 2009, the BEM-based FEM was introduced as an numerical approach for the treatment of boundary value problems. It is a Finite Element Method (FEM) that uses Trefftz-like basis functions which are defined to fulfil the underlying differential equation locally and which are treated by means of Boundary Element Methods (BEM). Due to the implicit definition of basis functions, this approach is applicable on general polygonal and polyhedral meshes and yields conforming approximations. The elements of the discretization do not necessarily have to be convex. After a review of the recent development of higher order basis functions the method is applied to a model problem on a sequence of meshes with L-shaped elements.

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A Non-standard Finite Element Method Based on Boundary Integral Operators

Cited by 18 publications

References 13 publications

Convection-adapted BEM-based FEM

Convection-adapted BEM-based FEM

Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

Higher order Trefftz‐like Finite Element Method on meshes with L‐shaped elements

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