Pith. Sign in

RECORD

Bernstein-Bezier Bases for Tetrahedral Finite Elements

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Mark Ainsworth, Guosheng Fu

Numerical Analysis · Submitted 27 Apr 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID1804.10466 (version 1)
Pith IDMRGLUDWU, Pith's permanent record of this paper and its review
CitationGuosheng Fu, Mark Ainsworth. "Bernstein-Bezier Bases for Tetrahedral Finite Elements." arXiv:1804.10466 (2018). https://pith.science/paper/MRGLUDWU
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,423 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1 -conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H1 -conforming Bernstein-Bezier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis. Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piece-wise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron. contribution 1.00 0.95 0.52 asserted 1.00
s1 The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1 -conforming Bernstein basis. method 0.64 0.86 0.10 asserted 1.00
s2 The basis functions respect the differential operators, namely, the gradients of the high-order H1 -conforming Bernstein-Bezier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis. contribution 0.80 0.87 0.32 asserted 1.00
s3 Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piece-wise) constant coefficients on affine tetrahedra. method 0.91 0.91 0.08 asserted 1.00
s4 Numerical results are presented to illustrate the use of the basis to approximate representative problems. result 0.79 0.94 0.09 asserted 0.98

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.61.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1 -conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H1 -conforming Bernstein-Bezier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis. Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piece-wise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.