Error estimates for generalized barycentric interpolation

Gillette, Andrew; Rand, Alexander; Bajaj, Chandrajit L.

doi:10.1007/s10444-011-9218-z

Cited by 62 publications

(84 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…A generic way of building reconstruction operators for any type of DoFs on polyhedral meshes has been proposed in Brezzi et al (2014), Christiansen (2008), Gillette et al (2014). The reconstruction operators are built locally in each mesh cell in such a way that suitable matching conditions are satisfied at mesh interfaces.…”

Section: Introductionmentioning

confidence: 99%

Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes

Bonelle¹,

Pietro

Ern

2015

Computer Aided Geometric Design

View full text Add to dashboard Cite

We study low-order reconstruction operators on polyhedral meshes, providing a unified framework for degrees of freedom attached to vertices, edges, faces, and cells. We present two equivalent sets of design properties and draw links with the literature. In particular, the two-level construction based on a P0-consistent and a stabilization part provides a systematic way of designing these operators. We present a simple example of piecewise constant reconstruction in each mesh cell, relying on geometric identities to fulfill the design properties on polyhedral meshes. Finally, we use these reconstruction operators to define a Hodge inner product and build Compatible Discrete Operator schemes, and we test the influence of the reconstruction operators in terms of accuracy and computational efficiency on an anisotropic diffusion problem

show abstract

Section: Introductionmentioning

confidence: 99%

Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes

Bonelle¹,

Pietro

Ern

2015

Computer Aided Geometric Design

View full text Add to dashboard Cite

show abstract

“…The first interpolates a given function u ∈ H 2 (Ω) using lower order trial functions. This operator has already been studied in [11] and some interpolation estimates have been shown. Additionally, we introduce two higher order interpolation operators and prove new estimates with similar but extended ideas.…”

Section: Interpolation Estimate and Fem Convergencementioning

confidence: 96%

“…Extensions to higher order. The lowest order harmonic trial functions are understood quite well in uniform [11] and adaptive [27] strategies. Therefore, the question for higher order approximations arises.…”

Section: Introductionmentioning

confidence: 98%

Higher Order BEM-Based FEM on Polygonal Meshes

Rjasanow¹,

Weißer²

2012

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

The BEM-based finite element method is reviewed and extended with higher order basis functions on general polygonal meshes. These functions are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are treated by means of boundary integral formulations and are approximated using the boundary element method in the numerics. To obtain higher order convergence, a new approximation of the material coefficient is proposed since previous strategies are not sufficient. Following recent ideas, error estimates are proved which guarantee quadratic convergence in the H 1 -norm and cubic convergence in the L 2 -norm. The numerical realization is discussed and several experiments confirm the theoretical results. Introduction.In the field of numerical methods for partial differential equations there is an increasing interest in nonsimplicial meshes. Several applications in solid mechanics, biomechanics, and geological science show a need for general elements within a finite element simulation. Discontinuous Galerkin methods [9] and mimetic finite difference methods [3] are able to handle such meshes. Nevertheless, these two strategies yield nonconforming approximations in their original versions which are not in the Sobolev space in which the exact solution lies.In 1975, Wachspress [26] proposed the construction of conforming rational basis functions on convex polygons with any number of sides. In recent years, several improved basis functions on polygonal elements have been introduced and applied in linear elasticity [24] or computer graphics [15,17], for example. There are even the first attempts which seek to introduce quadratic finite elements on polygons [20]. The recent publication [2] extends the nodal mimetic discretization strategy to arbitrary order on polygonal meshes.In [7] and the detailed work [6], a new kind of conforming finite element method on polygonal meshes has been proposed which uses basis functions that fulfill the differential equation locally. In the local problems constant material parameters and vanishing right-hand sides are prescribed. In case of the diffusion equation harmonic basis functions are recovered. This method has been studied in several articles concerning convergence [12,13] and residual error estimates for adaptive mesh refinement [27]. In the following, the theory is extended to a higher order method on polygonal meshes.The outline of this article is as follows. In section 2, some notation as well as a model problem are introduced. We review the first order method proposed in [7] and give a definition for regular meshes. The BEM-based FEM is extended to a higher order scheme in section 3. Afterward, we introduce interpolation operators in section 4 and prove interpolation properties which yield error estimates for the finite element

show abstract

“…In [11], the following error estimate with generalized barycentric coordinates interpolation I on polyhedra is established for Wachspress and Sibson coordinates, given sufficient geometric restrictions on the domain E; u − Iu H 1 (E) ≤ Ch|u| H 2 (E) , for all u ∈ H 2 (E). (16) If we interpolate h u h component-wise with Wachspress coordinates from vertex values reconstructed from the flux, Lemma 3 shows that this gives the reconstructed barycentric coordinate field R bci u F,h .…”

Section: Appendix: Proof Of Lemmamentioning

confidence: 99%

Velocity interpolation and streamline tracing on irregular geometries

2011

View full text Add to dashboard Cite

Streamline tracing on irregular grids requires reliable interpolation of velocity fields. We propose a new method for direct streamline tracing on polygon and polytope cells. While some numerical methods provide a basis function that can be used for interpolation, other methods provide only the fluxes at the faces of the elements. We introduce the concept of full-and rawfield methods. Full-field methods have built-in interpolation but are often not defined on general grids such as polygonal and polyhedral grids which we examine here. Also, reliability issues may arise on non-simplicial meshes in terms of not being able to reproduce constant velocity fields. We propose an interpolation in H(div) and H(curl) valid on general grids that is based on barycentric coordinates and that reproduces uniform flow. The interpolation can be used to compute the streamline directly on the complex cell geometry. The method generalizes to convex polytopes in 3D, with a restriction on the polytope topology near corners that is shown to be satisfied by several popular grid types. Numerical results confirm that the method is applicable to general grids and preserves uniform flow.

show abstract

Error estimates for generalized barycentric interpolation

Cited by 62 publications

References 38 publications

Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes

Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes

Higher Order BEM-Based FEM on Polygonal Meshes

Velocity interpolation and streamline tracing on irregular geometries

Contact Info

Product

Resources

About