A High-Order Lower-Triangular Pseudo-Mass Matrix for Explicit Time Advancement of hp Triangular Finite Element Methods

Brian Helenbrook; Jay Appleton

arxiv: 1906.10774 · v1 · pith:PS7O7GR6new · submitted 2019-06-25 · 🧮 math.AP · cs.NA· math.NA

A High-Order Lower-Triangular Pseudo-Mass Matrix for Explicit Time Advancement of hp Triangular Finite Element Methods

Jay Appleton , Brian Helenbrook This is my paper

Pith reviewed 2026-05-25 16:03 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA

keywords finite element methodstriangular elementsmass matrixexplicit time integrationpseudo-mass matrixhp finite elementsunstructured meshes

0 comments

The pith

No diagonal approximate mass matrix exists for accurate explicit time stepping on triangular elements.

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

The paper establishes that the standard polynomial space T(p) on triangles admits no diagonal approximate mass matrix capable of exactly projecting functions from the lower-degree space T(p-1). This property is required to preserve spatial accuracy during explicit time advancement. In its place the authors construct a lower-triangular pseudo-mass matrix together with a compatible high-order basis, and they verify the construction for the cubic case T(3). The resulting scheme permits efficient inversion-free time stepping on unstructured triangular meshes while retaining the full spatial accuracy of the underlying finite-element space.

Core claim

For the standard space of polynomials T(p) used with triangular elements, no diagonal approximate mass matrix yields an exact projection of functions in T(p-1). A lower-triangular pseudo-mass matrix combined with an accompanying high-order basis does permit such exact projections and therefore supports accurate explicit time advancement for T(3) on unstructured triangular meshes.

What carries the argument

Lower-triangular pseudo-mass matrix that replaces the standard (diagonal) approximate mass matrix while preserving exact projection onto T(p-1).

If this is right

Explicit time integration on triangular meshes can retain full spatial accuracy up to polynomial degree 3 without assembling or inverting a global mass matrix.
The same construction supplies a route to hp-adaptive explicit schemes on unstructured triangular grids.
Computational cost per time step becomes comparable to that of diagonal-mass quadrilateral spectral elements while retaining geometric flexibility.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The lower-triangular structure may extend to higher degrees p greater than 3, although the paper demonstrates only p=3.
Similar pseudo-mass constructions could be examined for other element shapes or mixed-element meshes where diagonal approximations also fail the projection test.
The method opens the possibility of comparing explicit triangular and quadrilateral schemes on the same geometry to quantify any remaining efficiency gap.

Load-bearing premise

Accuracy of an approximate mass matrix is defined by whether it produces an exact projection of all functions belonging to T(p-1).

What would settle it

An explicit diagonal matrix that, when applied to any function in T(p-1), recovers that function exactly after multiplication by the inverse of the true mass matrix, or a demonstration that the proposed lower-triangular matrix fails to recover T(2) functions exactly for T(3) elements.

read the original abstract

Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal, making it computationally efficient for explicit simulations. In this article it is shown that for the standard space of polynomials used with triangular elements, denoted $\mathcal{T}(p)$ where $p$ is the degree of the space, there is no diagonal approximate mass matrix that permits accurate solutions. Accuracy is defined as giving an exact projection of functions in $\mathcal{T}(p-1)$. In light of this, a lower-triangular pseudo-mass matrix method is introduced and demonstrated for the space $\mathcal{T}(3)$. The pseudo-mass matrix and accompanying high-order basis allow for computationally efficient time-stepping techniques without sacrificing the accuracy of the spatial approximation for unstructured triangular meshes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows no diagonal approximate mass matrix works for triangular T(p) under the exact T(p-1) projection rule and gives a lower-triangular construction for p=3.

read the letter

The main point is that diagonal pseudo-mass matrices, which work for quads and hexes, cannot satisfy the accuracy condition for triangular elements, so the authors build a lower-triangular version instead and demonstrate it for T(3). This lets explicit time stepping avoid full mass-matrix inversion on unstructured triangular meshes while keeping the stated projection property. The non-existence argument and the concrete lower-triangular construction are the actual new pieces. The paper does a clean job of stating the limitation that comes from the triangular polynomial space and of offering a workable replacement that stays computationally cheap. The approach is a direct response to a known practical issue in hp-FEM. The soft spot is the accuracy definition itself. Requiring exact projection onto every function in T(p-1) is what rules out diagonals, but the stress-test note is right that a milder consistency condition might still preserve formal order and could allow a diagonal matrix. The paper needs to show why this exact threshold is required to prevent order reduction in the semi-discrete system; without that link the non-existence result stays tied to the chosen bar rather than to a broader accuracy requirement. The demonstration is only for T(3), so readers will also want evidence on whether the method extends or how the basis is chosen to make the lower-triangular form work. The underlying finite-element reasoning looks consistent with standard theory and shows no circularity or invented entities. This paper is for people working on explicit time integration and mass-matrix approximations inside triangular hp-FEM. A reader who already cares about spectral-element techniques on unstructured meshes will find the construction useful. It deserves a serious referee because the claim is specific enough to check and the technique addresses a real gap even if the justification for the accuracy criterion needs tightening. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper shows that no diagonal approximate mass matrix exists for the standard polynomial space T(p) on triangles such that the mass matrix yields an exact projection of all functions in T(p-1). It then constructs a lower-triangular pseudo-mass matrix together with a compatible high-order basis for the specific case T(3), enabling explicit time advancement on unstructured triangular meshes without loss of spatial accuracy.

Significance. If the non-existence result holds under the stated accuracy definition and the T(3) construction is correct, the work supplies a practical route to explicit time marching for high-order triangular elements, filling a gap relative to the diagonal mass matrices available for quadrilaterals. The concrete demonstration for degree 3 is a tangible contribution.

major comments (1)

[Abstract and accuracy-definition section] Abstract and the section introducing the accuracy criterion: the non-existence claim for diagonal matrices is predicated on the requirement of exact projection onto all of T(p-1). The manuscript does not derive why this particular consistency condition is necessary and sufficient to prevent order reduction in the semi-discrete system under explicit time marching; a weaker condition (e.g., exactness against a proper subspace or quadrature exactness of degree 2p-2) might still permit a diagonal matrix while preserving formal order.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the single major comment below and indicate the corresponding revision.

read point-by-point responses

Referee: [Abstract and accuracy-definition section] Abstract and the section introducing the accuracy criterion: the non-existence claim for diagonal matrices is predicated on the requirement of exact projection onto all of T(p-1). The manuscript does not derive why this particular consistency condition is necessary and sufficient to prevent order reduction in the semi-discrete system under explicit time marching; a weaker condition (e.g., exactness against a proper subspace or quadrature exactness of degree 2p-2) might still permit a diagonal matrix while preserving formal order.

Authors: We agree that the original manuscript introduces the accuracy criterion (exact projection onto all of T(p-1)) without an explicit derivation of its relation to order preservation. This condition is the natural consistency requirement ensuring the approximate operator reproduces the exact mass matrix action on the subspace T(p-1), thereby preventing the introduction of lower-order spatial errors that would reduce the formal order of the semi-discrete system. In the revised manuscript we will add a short derivation in the accuracy-definition section that expands the error analysis for the semi-discrete hyperbolic problem and shows sufficiency of the stated condition for order retention. We note that the non-existence result is proved specifically under this condition; whether weaker conditions (such as exactness on a proper subspace or quadrature of degree 2p-2) admit a diagonal matrix is an interesting open question outside the scope of the present work. revision: yes

Circularity Check

0 steps flagged

No circularity: non-existence result follows from direct analysis of polynomial space T(p) under explicitly stated accuracy criterion

full rationale

The paper defines accuracy as exact projection of all functions in T(p-1), then shows no diagonal approximate mass matrix satisfies this property for the standard triangular polynomial space T(p). This is a direct algebraic or approximation-theoretic argument on the space itself rather than a fitted parameter, self-referential definition, or self-citation chain. The lower-triangular pseudo-mass construction for T(3) is presented as an alternative that meets the same criterion. No load-bearing step reduces by construction to its own inputs; the derivation remains self-contained against the chosen consistency condition. The appropriateness of the T(p-1) exactness threshold is a question of modeling assumptions, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the given text. The pseudo-mass matrix is a constructed numerical object whose internal details are not supplied.

pith-pipeline@v0.9.0 · 5683 in / 1254 out tokens · 56629 ms · 2026-05-25T16:03:41.781288+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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R. Pasquette and F. Rapetti, Cubature versus fekete-gauss nodes for spectral element meth- ods on simplicial meshes , Journal of Computational Physics, 347 (2017), pp. 463–466

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M. D. Samson, H. Li, and L.-L. Wang , A new triangular spectral element method I: Im- plementation and analysis on a triangle , Numerical Algorithms, 64 (2012), pp. 519–547, http://dblp.uni-trier.de/db/journals/na/na64.html#SamsonLW13

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S. Sherwin and G. Karniadakis , A triangular spectral element method; applications to the incompressible Navier-Stokes equations , Computer Methods in Applied Mechanics and Engineering, 123 (1995), pp. 189–229

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Sousa and U

E. Sousa and U. Shumlak , A blended continuous-discontinuous ﬁnite element method for solving the multi-ﬂuid plasma model, Journal of Computational Physics, 326 (2016), pp. 56– 75

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J. Sun, K. Lee, and L. H.P. , Comparison of implicit and explicit ﬁnite element methods for dynamic problems, Journal of Materials Processing Technology, 105 (2000), pp. 110–118

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M. A. Taylor and B. Wingate, A generalized diagonal mass matrix spectral element method for non-quadrilateral elements , Applied Numerical Mathematics, 33 (2000), pp. 259–265

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[28]

Zarei, B

S. Zarei, B. Oskooi, N. Amini, and A. R. Dalkhani , 2D spectral element modeling of GPR wave propagation in inhomogeneous media , Journal of Applied Geophysics, 133 (2016), pp. 92–97

work page 2016

[1] [1]

Abgrall, Higher order schemes for hyperbolic problems using globally continuous approxi- mation and avoiding mass matrices , Journal of Scientiﬁc Computing, (2017)

R. Abgrall, Higher order schemes for hyperbolic problems using globally continuous approxi- mation and avoiding mass matrices , Journal of Scientiﬁc Computing, (2017)

work page 2017

[2] [2]

Abgrall, Q

R. Abgrall, Q. Viville, H. Beaugendre, and C. Dobrzynski, Construction of a p-adaptive continuous residual distribution scheme , Journal of Scientiﬁc Computing, 72 (2017), pp. 1232–1268

work page 2017

[3] [3]

M. G. Armentano and R. G. Dur ´an, Mass-lumping or not mass-lumping for eigenvalue problems, Numerical Methods for Partial Diﬀerential Equations, 19 (2003), pp. 653–664

work page 2003

[4] [4]

M. J. Brazell and B. T. Helenbrook , p = 2 continuous ﬁnite elements on tetrahedra with local mass matrix inverstion to solve the preconditioned compressible Navier-Stokes equa- tions, Computers and Fluids, 88 (2013), pp. 642–652

work page 2013

[5] [5]

Canuto, M

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang , Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics , Springer, 2007

work page 2007

[6] [6]

L. Chen, J. Shen, and C. Xu , A triangular spectral method for the Stokes equations , Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 158–179

work page 2011

[7] [7]

Chin-Joe-Kong, W

M. Chin-Joe-Kong, W. A. Mulder, and M. Van Veldhuizen , Higher-order triangular and tetrahedral ﬁnite elements with mass lumping for solving the wave equation , Journal of Engineering Mathematics, 35 (1999), pp. 405–426

work page 1999

[8] [8]

Cohen, P

G. Cohen, P. Joly, J. Roberts, and N. Tordjman , Higher order triangular ﬁnite elements with mass lumping for the wave equation, SIAM J. Numer. Anal., 38 (2001), pp. 2047–2078

work page 2001

[9] [9]

Dubiner, Spectral methods on triangles and other domains, Journal of Scientiﬁc Computing, 6 (1991), pp

M. Dubiner, Spectral methods on triangles and other domains, Journal of Scientiﬁc Computing, 6 (1991), pp. 345–390

work page 1991

[10] [10]

Durufle, P

M. Durufle, P. Grob, and P. Joly , Inﬂuence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral ﬁnite element method in the time domain , Numerical Methods for Partial Diﬀerential Equations, 25 (2009), pp. 526–551

work page 2009

[11] [11]

Eskilsson and S

C. Eskilsson and S. Sherwin , A triangular spectral/ hp discontinuous Galerkin method for modelling 2d shallow water equations , International Journal for Numerical Methods in Fluids, 45 (2004), pp. 605–623

work page 2004

[12] [12]

Fi´etier and M

N. Fi´etier and M. O. Deville , Simulations of time-dependent ﬂows of viscoelastic ﬂuid with spectral element methods, Journal of Scientiﬁc Computing, (2002)

work page 2002

[13] [13]

Franca and A

L. Franca and A. Russo , Mass lumping emanating from residual-free bubbles , Computer Methods in Applied Mechanics and Engineering, 142 (1997), pp. 353–360. 20 JAY M. APPLETON AND B. T. HELENBROOK

work page 1997

[14] [14]

Giraldo and T

F. Giraldo and T. Warburton, A nodal triangle-based spectral method for the shallow water equations on the sphere , Journal of Computational Physics, 207 (2005), pp. 129–150

work page 2005

[15] [15]

F. X. Giraldo and M. A. Taylor, A diagonal mass matrix triangular spectral element method based on cubature points, Journal of Engineering Mathematics, 56 (2006), pp. 307–322

work page 2006

[16] [16]

B. T. Helenbrook, On the existence of explicit hp-ﬁnite element methods using Gauss-Lobatto integration on the triangle , SIAM J. Numer. Anal., 47 (2009), pp. 1304–1318

work page 2009

[17] [17]

J. S. Hesthaven, From electrostatic to almost optimal nodal sets for polynomial interpolation in a simplex , SIAM J. NUMER. ANAL., 35 (1998), pp. 655–676

work page 1998

[18] [18]

J. S. Hesthaven, Spectral penalty methods, Applied Numerical Mathematics, 33 (2000), pp. 23– 41

work page 2000

[19] [19]

Karniadakis and S

G. Karniadakis and S. Sherwin , Spectral/hp element methods for computational ﬂuid dy- namics, in Numerical Mathematics and Scientiﬁc Computation, G. Golub, A. Greenbaum, A. Stuart, and E. S¨ uli, eds., Oxford Science Publications, 2 ed., 2005

work page 2005

[20] [20]

Mavriplis, Unstructured grid techniques, Annu

D. Mavriplis, Unstructured grid techniques, Annu. Rev. Fluid. Mech., 29 (1997), pp. 473–514

work page 1997

[21] [21]

Nicolas, L

A. Nicolas, L. Nicolas, and C. Vollaire , An explicit 2D ﬁnite element time domain scheme for electromagnetic wave propagation, IEEE Transactions on Magnetics, 35 (1999), pp. 1538–1541

work page 1999

[22] [22]

Pasquette and F

R. Pasquette and F. Rapetti, Cubature versus fekete-gauss nodes for spectral element meth- ods on simplicial meshes , Journal of Computational Physics, 347 (2017), pp. 463–466

work page 2017

[23] [23]

M. D. Samson, H. Li, and L.-L. Wang , A new triangular spectral element method I: Im- plementation and analysis on a triangle , Numerical Algorithms, 64 (2012), pp. 519–547, http://dblp.uni-trier.de/db/journals/na/na64.html#SamsonLW13

work page 2012

[24] [24]

Sherwin and G

S. Sherwin and G. Karniadakis , A triangular spectral element method; applications to the incompressible Navier-Stokes equations , Computer Methods in Applied Mechanics and Engineering, 123 (1995), pp. 189–229

work page 1995

[25] [25]

Sousa and U

E. Sousa and U. Shumlak , A blended continuous-discontinuous ﬁnite element method for solving the multi-ﬂuid plasma model, Journal of Computational Physics, 326 (2016), pp. 56– 75

work page 2016

[26] [26]

J. Sun, K. Lee, and L. H.P. , Comparison of implicit and explicit ﬁnite element methods for dynamic problems, Journal of Materials Processing Technology, 105 (2000), pp. 110–118

work page 2000

[27] [27]

M. A. Taylor and B. Wingate, A generalized diagonal mass matrix spectral element method for non-quadrilateral elements , Applied Numerical Mathematics, 33 (2000), pp. 259–265

work page 2000

[28] [28]

Zarei, B

S. Zarei, B. Oskooi, N. Amini, and A. R. Dalkhani , 2D spectral element modeling of GPR wave propagation in inhomogeneous media , Journal of Applied Geophysics, 133 (2016), pp. 92–97

work page 2016