Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption Part 1: Lagrange interpolation on triangles

Kenta Kobayashi; Takuya Tsuchiya

REVIEW 3 major objections 6 minor 28 references

Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption Part 1: Lagrange interpolation on triangles

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every triangle, the Lagrange interpolation error is controlled by the circumradius alone, with no shape-regularity constant.

desk verdict A mostly clean lecture write-up of a genuinely useful circumradius-based interpolation estimate, with one real gap in Lemma 23 and p=∞ left as an exercise. read the letter →

arxiv 1908.03894 v3 pith:GJLGL7ED submitted 2019-08-11 math.NA cs.NA

Kenta Kobayashi , Takuya Tsuchiya This is my paper · ORCID

classification math.NAcs.NA MSC 65N1565N3041A0541A10

keywords Lagrangeinterpolationfiniteelementerroranalysiscircumradiusconditionshaperegularitymaximumangleanisotropictriangulationsqueezingtransformationdifferencequotients

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This lecture-style paper removes the shape-regularity assumption from the standard error analysis of Lagrange interpolation on triangles. It proves that for any triangle $K$, degree $k$, and $1\le p\le\infty$, the interpolation error in the $W^{m,p}$ seminorm is controlled by the circumradius $R_K$ rather than by ratios of edge lengths or angles: $|v-I_k^K v|_{m,p,K} \le C_{k,m,p}(R_K/h_K)^m h_K^{k+1-m}|v|_{k+1,p,K}$. The constant depends only on $k$, $m$, and $p$, not on how thin or degenerate the triangle is. This matters because adaptive meshes can contain arbitrarily flat elements, and the result says convergence of finite element solutions is still guaranteed once the circumradius of every element tends to zero at the right rate.

What carries the argument

The load-bearing construction is the squeezing argument combined with a two-factor decomposition of the affine map from the reference triangle: $A = \tilde A D_{\alpha\beta}$, where $D_{\alpha\beta}=\mathrm{diag}(\alpha,\beta)$ is a diagonal scaling and $\tilde A$ has determinant one. The diagonal scaling produces the squeezed triangle $K_{\alpha\beta}$; the proof shows that interpolation constants on $K_{\alpha\beta}$ are bounded by $(\max\{\alpha,\beta\})^{k+1-m}$ times a universal constant. The remaining unimodular factor $\tilde A$ contributes at most a factor $(R_K/h_K)^m$. To control the universal constant, the paper introduces grid difference quotients: functions vanishing at all Lagrange nodes have derivatives whose integrals over certain grid rectangles $\square^\delta_{lp}$ vanish, and these integral conditions are linearly independent on the relevant polynomial space, yielding finite constants $A^{\gamma,k}_p$.

What would settle it

Run a computer-algebra search over $k\ge 4$ for a nonzero polynomial $q\in P_{k-|\delta|}$ satisfying $\int_{\square^\delta_{lp}} q = 0$ for every grid rectangle or segment $\square^\delta_{lp}$ in the reference triangle $\hat K$; any such $q$ would make Lemma 23 false and $A^{\gamma,k}_p$ infinite, breaking the proof of the circumradius estimate.

Watch

Extended reading notes

Core claim

The central claim is the circumradius estimate (Theorem 10, restated as Corollary 28): for an arbitrary triangle $K$ with circumradius $R_K$ and diameter $h_K$, for Lagrange interpolation $I_k^K$ of degree $k$, and for $1\le p\le\infty$, $0\le m\le k$, one has $|v-I_k^K v|_{m,p,K} \le C_{k,m,p}(R_K/h_K)^m h_K^{k+1-m}|v|_{k+1,p,K}$ for every $v\in W^{k+1,p}(K)$, with $C_{k,m,p}$ depending only on $k$, $m$, and $p$. The dimensionless ratio $R_K/h_K$ equals $1/(2\sin\theta_{\max})$, where $\theta_{\max}$ is the largest angle, so the bound interpolates between the classical maximum-angle regime and genuinely degenerate triangles. In particular, when $R_K$ tends to zero, the interpolation error tends to zero even if the triangle collapses toward a segment.

Load-bearing premise

The bound rests on the assertion that for every interpolation order $k$, no nonzero polynomial of degree no greater than $k-|\gamma|$ integrates to zero over all the relevant grid rectangles; the paper checks only $k=2$ and $k=3$ and asserts the general case, so if that linear-independence claim fails the whole estimate collapses.

Editorial extensions

If this is right

For piecewise linear elements on a polygonal domain, the $H^1$ error of the Poisson finite element solution is bounded by $C\max_{K\in T_h} R_K |u|_{2,2,\Omega}$, so convergence follows as soon as all element circumradii go to zero, no matter how sharp the angles are.
Higher-order elements can repair bad triangulations: for degree $k\ge 2$ the factor becomes $R_K h_K^{k-1}$, so even families of triangles with $R_K=O(1)$ converge as $h\to 0$, albeit with a possibly reduced rate.
When $R_K/h_K$ is bounded, the estimate reduces to the classical $O(h_K^{k+1-m})$ bound and reproduces the maximum-angle/semiregularity regime as a special case.
The result is geometrically scale-consistent: refining any fixed skinny triangle preserves the estimate because both sides scale like the appropriate power of $h_K$, with no hidden aspect-ratio constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper treats two-dimensional triangles only, despite its title mentioning simplicial triangulations; extending the difference-quotient integral conditions to tetrahedral lattices would be a natural test of whether the circumradius remains the governing quantity in 3D.
The asserted linear independence in Lemma 23 is stated for general $k$ but verified only for $k=2,3$; a computer-algebra check for $k=4,5$ would confirm whether the general claim is true and would also complete the missing $p=\infty$ case by the same contradiction argument.
The numerical experiments plot $H^1$ error against maximum circumradius and show nearly identical convergence rates across very different triangle families; the same data-collapse test could be applied to $L^2$ errors or to adaptive refinement loops to see whether circumradius is the right mesh-quality monitor beyond the specific problem treated here.
If the bound is sharp, anisotropic mesh adaptation should target element circumradius rather than inradius or edge ratio when guaranteeing interpolation accuracy.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is not new research, and the authors do not claim it is. It is exactly what it says it is—a plain account of why the circumradius, not shape regularity or maximum angle, controls Lagrange interpolation error on triangles. I think it earns its place as a useful reference for graduate students, but there is one place where the proof needs work before I would point students at it as self-contained.

The good parts first. The main message is real: for an arbitrary triangle, the bound is (R_K/h_K)^m h_K^{k+1-m}, and the linear case with constant 1 (Corollary 9) is a nice, nontrivial result. The paper does a good job explaining the standard shape-regularity analysis, the Babuška–Aziz squeezing idea, and how R_K/h_K emerges from the norm of the matrix tilde A. The examples in Section 1 and the numerical experiments in Section 8 line up with the theory. It is honest exposition, and the didactic choices—deferring some details to exercises—are mostly fine.

Now the soft spots, in proportion. Lemma 23 is load-bearing: it says the difference-quotient integral conditions on the rectangles are linearly independent on P_{k-|γ|}. The proof gives a dimension count and examples for k=2,3, then asserts the general case. Dimension counting does not imply injectivity of a square linear system, and Lemma 24 depends on this exactly. If the claim failed for some k, the constants A^p_{γ,k} in (32) could be infinite and the squeezing argument would collapse. I do not think the claim is false—it is presumably proven in the authors' earlier papers—but this manuscript does not give the general argument or point to a specific lemma where it is done. A referee should ask for that. Also, the p=∞ case of Theorem 21 is left as an exercise, so as written the theorem statement for 1≤p≤∞ is not fully proven. And Corollary 28 says v∈W^{2,p}(K) where it should be W^{k+1,p}(K); that is a typo, not a mathematical issue.

Is this worth refereeing? Yes, if the venue takes expository papers seriously. The subject is important, the exposition is mostly very good, and the gap is fixable. If it is aimed at a research journal, I would insist on the Lemma 23 fix and a proof or explicit citation for p=∞ before acceptance. For your own use, I would keep a copy for the classroom, but I would not cite it as a primary source—I would cite the original papers.

Referee Report

3 major / 6 minor

Summary. The manuscript is an expository lecture note whose central object is the error analysis of k-th order Lagrange interpolation on arbitrary triangles in R^2. The authors aim to prove that for 1≤p≤∞, 0≤m≤k, and any triangle K with circumradius R_K, the estimate |v - I_k^K v|_{m,p,K} ≤ C_{k,m,p} (R_K/h_K)^m h_K^{k+1-m} |v|_{k+1,p,K} holds for all v in W^{k+1,p}(K), with C depending only on k, m, and p. The proof strategy is to bound interpolation on right triangles K_{αβ} squeezed from the reference triangle using Babuška–Aziz type constants, to extend this to arbitrary triangles through the decomposition A = ~A D_{αβ}, and then to estimate the norm of ~A and its inverse in terms of R_K/h_K. Numerical experiments in Section 8 support the estimate and illustrate that convergence can fail when measured against h but hold when measured against the circumradius.

Significance. If completed, the main estimate is significant: it replaces the shape-regularity or maximum-angle condition by the circumradius as the controlling geometric quantity, with direct consequences for adaptive and anisotropic meshes. The paper is self-contained in its classical parts, gives explicit k=1 constants in Corollary 9, and includes numerical experiments that clearly separate h-dependence from R-dependence. The proof strategy via squeezing transformations and difference quotients is elegant. However, the manuscript is explicitly presented as a lecture note rather than as a new research contribution, and its value is primarily pedagogical; the central result is a re-presentation of the authors' earlier results in [15,17,18].

major comments (3)

[Section 6, Lemma 23] The proof of Lemma 23 is incomplete. The assertion that the integral conditions on the rectangles □^γ_{lp} 'are linearly independent and determine q = 0 uniquely' is exactly the injectivity claim that needs proof; the preceding dimension count dim P_{k-|δ|} = #{□^δ_{lp} ⊂ \hat K} only shows that the linear system is square. Only the cases k=2 and k=3 are illustrated. Since Lemma 24 relies on Lemma 23 to conclude that the limit polynomial \bar q is zero, and Theorem 21 relies on Lemma 24, the central estimate Corollary 28 rests on this unproved combinatorial statement. Please provide a proof for general k and γ, or give a precise reference to a proof.
[Section 6, proof of Theorem 21] Theorem 21 is stated for 1 ≤ p ≤ ∞, but the proof in Section 6 covers only 1 ≤ p < ∞; the p = ∞ case is relegated to an exercise immediately after Lemma 24. Because Corollary 22 and ultimately Corollary 28 inherit the full range 1 ≤ p ≤ ∞, the manuscript as written does not prove the main theorem for p = ∞. Please add the short max-norm argument or restrict the statement of the theorem to 1 ≤ p < ∞.
[Sections 5.2 and 6, notation for rectangles] The notation for the rectangles is a source of avoidable confusion. In (35) the symbol □^{(t,s)}_γ denotes a rectangle with displacement (t,s) and lower-left corner x_γ, while in Section 6 the same symbol □^γ_{lp} is reused with γ standing for the derivative multi-index in (30). The implication (31) depends on identifying these two roles. Please define explicitly that □^γ_{lp} is the rectangle with displacement γ and lower-left corner (l/k, p/k), and state why (35) yields (31) for every such rectangle.

minor comments (6)

[Corollary 28] The statement says 'for any v ∈ W^{2,p}(K)', which is inconsistent with the presence of the semi-norm |v|_{k+1,p,K} on the right-hand side; it should read W^{k+1,p}(K). In addition, one occurrence of the factor writes (R_K/h_k)^m instead of (R_K/h_K)^m.
[Corollary 26] The last semi-norm is written as |v|_{k+1,p,T}; the domain should be K, not T.
[Section 8] The text says 'We replot the same data in Figure 4', but the figure being referred to is Figure 8; the cross-reference should be corrected.
[Section 2.5] 'Morry's inequality' should be 'Morrey's inequality'.
[Throughout] There are several typographical slips, including 'C_{k,m.p}' for 'C_{k,m,p}' in Lemma 25 and Corollary 26, and 'intergers' for 'integers' in Corollary 28.
[Theorem 14 proof] In the contradiction argument the expression 'lim_{lk→0}' should presumably be 'lim_{l_m→∞}'; the current notation is confusing.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the circumradius estimate is derived self-contained; only proof-completeness gaps (Lemma 23, p=∞ exercise) are noted.

full rationale

The central estimate (Corollary 28 / Theorem 10) is obtained by a self-contained chain: Theorem 15/16 gives the standard affine-transformation error bound; the squeezing Theorem 21 is proved in Sections 4-6 by defining the difference-quotient spaces Ξ^{γ,k}_p and proving finiteness of A^{γ,k}_p by contradiction with Ciarlet's Theorem 14 and Gagliardo-Nirenberg; Lemma 25 transfers the bound to a general triangle via A = ÃD_{αβ}; and (37) bounds ‖Ã^{-1}‖ by C R_K/h_K. The constant C_{k,m,p} is shown to exist by compactness, not fitted to data, and the circumradius enters through the derived inequality (37), not through an assumed estimate. Self-citations [14,15,17,18] are used for motivation, context, and prior statements, but every load-bearing lemma used here (Lemmas 17-19, 24, 25 and Theorem 21 except for the p=∞ case assigned as an exercise) is proved in the manuscript with external tools. The only weaknesses are completeness gaps: Lemma 23's linear-independence of the rectangle integral conditions is asserted after k=2,3 examples rather than proved for all k, and the p=∞ part of Theorem 21 is left to Exercise; these affect whether the proof as written is complete, but they are not circular reductions of the target estimate to itself.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim does not require free parameters fitted to data. It relies on standard Sobolev space tools and on the paper's own constraint spaces, whose finiteness is proven modulo a sketched combinatorial lemma. The main unproved background assumptions are the standard imbedding and compactness theorems used throughout.

assumptions (4)

standard math Sobolev imbedding theorem (e.g., W^{2,p}(K) ⊂ C^0(K) for 1≤p≤∞)
Invoked in Section 2.5 to ensure point values of v∈W^{k+1,p}(K) are well-defined so Lagrange interpolation is valid.
standard math Gagliardo-Nirenberg inequality (Theorem 13)
Used in Lemma 19 and Lemma 24 to bound intermediate seminorms from |u|_{0,p} and the top seminorm on the reference triangle.
standard math Ciarlet's quotient norm equivalence (Theorem 14)
Used in Theorem 15 and Lemma 24; relies on compactness of W^{k+1,p}(Ω) into W^{k,p}(Ω) and Hahn-Banach extension of dual basis functions.
ad hoc to paper Lemma 23: integral conditions over difference-quotient rectangles are linearly independent on P_{k-|γ|}
Asserted in Section 6 with only examples for k=2,3; the general proof is omitted. This ensures A_{γ,k}^p<∞ in Lemma 24 and is load-bearing for Theorem 21.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption Part 1: Lagrange interpolation on triangles." pith.science (2026). https://pith.science/paper/GJLGL7ED

@misc{pith2026190803894,
  author       = {Pith},
  title        = {Pith review of: Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption Part 1: Lagrange interpolation on triangles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJLGL7ED}},
  note         = {Machine review of arXiv:1908.03894}
}

read the original abstract

In the error analysis of finite element methods, the shape regularity assumption on triangulations is typically imposed to obtain a priori error estimations. In practical computations, however, very thin or degenerated elements that violate the shape regularity assumption may appear when we use adaptive mesh refinement. In this manuscript, we attempt to establish an error analysis approach without the shape regularity assumption on triangulations. We have presented several papers on the error analysis of finite element methods on non-shape regular triangulations. The main points in these papers are that, in the error estimates of finite element methods, the circumradius of the triangles is one of the most important factors. The purpose of this manuscript is to provide a simple and plain explanation of the results to researchers and, in particular, graduate students who are interested in the subject. Therefore, this manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.

Figures

Figures reproduced from arXiv: 1908.03894 by the authors.

**Figure 1.** Set Σk (K), k = 1, k = 2, k = 3. Let Pk(K) be a set of polynomials defined on K whose degree is at most k. For a continuous function v ∈ C 0 (K), the kth-order Lagrange interpolation I k Kv ∈ Pk(K) is defined as v(x) = (I k Kv)(x), ∀x ∈ Σ k (K). To enable the error analysis of Lagrange interpolation, we typically introduce the following condition [8, 6, 10]. Let hK := diamK and ρK be the diameter of its inscribed ci… view at source ↗

**Figure 2.** General triangle K in the standard position. The vertices are x1 = (0, 0)>, x2 = (α, 0)>, and x3 = (βs, βt) >, where s 2 + t 2 = 1, t > 0. We assume that 0 < β ≤ α ≤ hK. These assumptions imply that the affine transformation ϕ can be written as ϕ(x) = Ax with the matrix A = α βs 0 βt . (3) 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Examples of triangles that violate the maximum angle condition but satisfy [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Squeezing the reference triangle Kb perpendicularly does not deteriorate the approximation property of Lagrange interpolation. Applying Theorem 21 to v − Ik Kαβ v ∈ T k p (Kαβ) for v ∈ Wk+1,p(Kαβ), and obtain the following corollary. Corollary 22 For arbitrary v ∈ Wk+1…

**Figure 5.** Figure 5: The six squares of size 1/4 for δ = (1, 1) and the (union of) six segments of length 1/2 for δ = (2, 0) in Kb. are equal to 0, then we have a = b = c = 0, that is, q(x, y) = 0. The case γ = (0, 1) is similar. Let k = 3 and γ = (1, 0). Then, k−|γ| = 2. Set q(x, y) = a+b…

**Figure 6.** Figure 6: The standard position of a general triangle (reprint). The vertices are [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]

**Figure 7.** Figure 7: Triangulation of Ω with N = 12 and α = 1.6 and the errors for FEM solutions in the H1 -norm. The horizontal axis represents the maximum diameter of the triangles and the vertical axis represents the H1 -norm of the errors of the FEM solutions. The number next to the sy…

**Figure 8.** Figure 8: Replotted data: the errors in the H1 -norm of FEM solutions measured using the circumradius. The horizontal axis represents the maximum circumradius of the triangles. Acknowledgments We thank Dr. Th´eophile Chaumont-Frelet for his valuable comments. References [1] R.A.…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Adams, J.J.F

R.A. Adams, J.J.F. Fournier: Sobolev Spaces, 2nd edition, Pure and Applied Math- ematics 140, Elsevier/Academic Press, New York, 2003

work page 2003
[2]

Apel : Anisotropic Finite Element: Local estimates and applications, Advances in Numerical Mathematics

T. Apel : Anisotropic Finite Element: Local estimates and applications, Advances in Numerical Mathematics. B.G. Teubner, Stuttgart, 1999

work page 1999
[3]

Atkinson: An Introduction to Numerical Analysis, 2nd edition, John Wiley & Sons, New York, 1989

K.E. Atkinson: An Introduction to Numerical Analysis, 2nd edition, John Wiley & Sons, New York, 1989

work page 1989
[4]

Babuˇ ska, A.K

I. Babuˇ ska, A.K. Aziz: On the angle condition in the ﬁnite element method, SIAM J. Numer. Anal. 13 (1976), 214–226

work page 1976
[5]

Barnhill, J.A

R.E. Barnhill, J.A. Gregory : Sard kernel theorems on triangular domains with application to ﬁnite element error bounds. Numer. Math., 25 (1976), 215-229

work page 1976
[6]

Brenner, L.R

S.C. Brenner, L.R. Scott : The Mathematical Theory of Finite Element Methods. 3rd edition. Texts in Applied Mathematics 15, Springer, New York, 2008

work page 2008
[7]

Brezis: Functional Analysis, Sobolev Spaces and Partial Diﬀerential Equations

H. Brezis: Functional Analysis, Sobolev Spaces and Partial Diﬀerential Equations. Universitext, Springer, New York, 2011. 34

work page 2011
[8]

Ciarlet: The Finite Element Methods for Elliptic Problems

P.G. Ciarlet: The Finite Element Methods for Elliptic Problems. Classics in Applied Mathematics 40, SIAM, Philadelphia, 2002, Reprint of the 1978 original (North Holland, Amsterdam)

work page 2002

Show all 28 references

[9]

Brandts, S

J. Brandts, S. Korotov, M. Kˇ r´ ıˇ zek: On the equivalence of regularity criteria for tri- angular and tetrahedral ﬁnite element partitions, Comput. Math. Appl. 55 (2008), 2227–2233

2008
[10]

Ern, J-L

A. Ern, J-L. Guermond : Theory and Practice of Finite Elements. Applied Mathe- matical Sciences 159, Springer, New York, 2004

2004
[11]

Hannukainen, S

A. Hannukainen, S. Korotov, M. Kˇ r´ ıˇ zek: The maximum angle condition is not necessary for convergence of the ﬁnite element method, Numer. Math., 120, 79–88 (2012)

2012
[12]

Horn, C.R

R.A. Horn, C.R. Johnson : Topics in Matrix Analysis, Cambridge Univ. Press, 1991

1991
[13]

Jamet : Estimations d’erreur pour des elements ﬁnis droits presque degeneres

P. Jamet : Estimations d’erreur pour des elements ﬁnis droits presque degeneres. R.A.I.R.O. Anal. Numer., 10 (1976), 43–61

1976
[14]

Kobayashi : On the interpolation constants over triangular elements (in Japanese), RIMS Kokyuroku, 1733 (2011), 58-77

K. Kobayashi : On the interpolation constants over triangular elements (in Japanese), RIMS Kokyuroku, 1733 (2011), 58-77

2011
[15]

Kobayashi, T

K. Kobayashi, T. Tsuchiya : A Babuˇ ska-Aziz type proof of the circumradius condi- tion, Japan J. Indust. Appl. Math., 31 (2014), 193-210

2014
[16]

Kobayashi, T

K. Kobayashi, T. Tsuchiya : On the circumradius condition for piecewise linear triangular elements, Japan J. Indust. Appl. Math., 32 (2015), 65–76

2015
[17]

Kobayashi, T

K. Kobayashi, T. Tsuchiya : A priori error estimates for Lagrange interpolation on triangles. Appl. Math., Praha 60 (2015), 485–499

2015
[18]

Math., Praha 61 (2016), 121–133

K.Kobayashi, T.Tsuchiya : Extending Babuˇ ska-Aziz’s theorem to higher order La- grange interpolation, Appl. Math., Praha 61 (2016), 121–133

2016
[19]

Kˇ r´ ıˇ zek: On semiregular families of triangulations and linear interpolation

M. Kˇ r´ ıˇ zek: On semiregular families of triangulations and linear interpolation. Appl. Math., Praha 36 (1991), 223–232

1991
[20]

Kˇ r´ ıˇ zek: On the maximum angle condition for linear tetrahedral elements

M. Kˇ r´ ıˇ zek: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal., 29 (1992), 513–520

1992
[21]

Kufner, O

A. Kufner, O. John, S. Fuˇ c´ ık: Function Spaces. Noordhoﬀ International Publishing, Leyden, 1977. 35

1977
[22]

X. Liu, F. Kikuchi : Analysis and estimation of error constants for P0 and P1 in- terpolation over triangular ﬁnite element, J. Math. Sci. Univ. Tokyo, 17 (2010), 27–78

2010
[23]

Oswald : Divergence of fem: Babuˇ ska-Aziz triangulations revisited, Appl

P. Oswald : Divergence of fem: Babuˇ ska-Aziz triangulations revisited, Appl. Math., Praha 60 (2015), 473–484

2015
[24]

Shenk : Uniform error estimates for certain narrow Lagrange ﬁnite elements

N.A. Shenk : Uniform error estimates for certain narrow Lagrange ﬁnite elements. Math. Comp., 63 (1994), 105–119

1994
[25]

Yamamoto : Introduction to Numerical Analysis, 2nd edition, (In Japanese), Saiensu-sha, 2003

T. Yamamoto : Introduction to Numerical Analysis, 2nd edition, (In Japanese), Saiensu-sha, 2003

2003
[26]

Yamamoto: Elements of Matrix Analysis (in Japanese), Saiensu-sha, 2010

T. Yamamoto: Elements of Matrix Analysis (in Japanese), Saiensu-sha, 2010

2010
[27]

ˇZen´ ıˇ sek: The convergence of the ﬁnite element method for boundary value prob- lems of a system of elliptic equations

A. ˇZen´ ıˇ sek: The convergence of the ﬁnite element method for boundary value prob- lems of a system of elliptic equations. (in Czech) Appl. Math., 14 (1969), 355–377

1969
[28]

Zl´ amal: On the ﬁnite element method, Numer

M. Zl´ amal: On the ﬁnite element method, Numer. Math. 12 (1968), 394–409. 36

1968

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Adams, J.J.F

R.A. Adams, J.J.F. Fournier: Sobolev Spaces, 2nd edition, Pure and Applied Math- ematics 140, Elsevier/Academic Press, New York, 2003

work page 2003

[2] [2]

Apel : Anisotropic Finite Element: Local estimates and applications, Advances in Numerical Mathematics

T. Apel : Anisotropic Finite Element: Local estimates and applications, Advances in Numerical Mathematics. B.G. Teubner, Stuttgart, 1999

work page 1999

[3] [3]

Atkinson: An Introduction to Numerical Analysis, 2nd edition, John Wiley & Sons, New York, 1989

K.E. Atkinson: An Introduction to Numerical Analysis, 2nd edition, John Wiley & Sons, New York, 1989

work page 1989

[4] [4]

Babuˇ ska, A.K

I. Babuˇ ska, A.K. Aziz: On the angle condition in the ﬁnite element method, SIAM J. Numer. Anal. 13 (1976), 214–226

work page 1976

[5] [5]

Barnhill, J.A

R.E. Barnhill, J.A. Gregory : Sard kernel theorems on triangular domains with application to ﬁnite element error bounds. Numer. Math., 25 (1976), 215-229

work page 1976

[6] [6]

Brenner, L.R

S.C. Brenner, L.R. Scott : The Mathematical Theory of Finite Element Methods. 3rd edition. Texts in Applied Mathematics 15, Springer, New York, 2008

work page 2008

[7] [7]

Brezis: Functional Analysis, Sobolev Spaces and Partial Diﬀerential Equations

H. Brezis: Functional Analysis, Sobolev Spaces and Partial Diﬀerential Equations. Universitext, Springer, New York, 2011. 34

work page 2011

[8] [8]

Ciarlet: The Finite Element Methods for Elliptic Problems

P.G. Ciarlet: The Finite Element Methods for Elliptic Problems. Classics in Applied Mathematics 40, SIAM, Philadelphia, 2002, Reprint of the 1978 original (North Holland, Amsterdam)

work page 2002