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T-IFISS: a toolbox for adaptive FEM computation

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Pith's one-line read T-IFISS implements self-adaptive finite element workflows with rigorous a posteriori error control for deterministic and parametric elliptic PDEs.

desk verdict A genuinely useful, well-documented adaptive FEM toolbox; algorithms are prior work but the integration is new, and the only real weakness is the unquantified approximate reference solutions in the effectivity plots. read the letter →

arxiv 1908.05618 v2 pith:XDCNAPZD submitted 2019-08-15 math.NA cs.NA

Alex Bespalov , Leonardo Rocchi , David Silvester This is my paper · ORCID
classification math.NAcs.NA MSC 97N8065N3065N1565N5035R6065C2065N22
keywords finiteelementsadaptivemethodsaposteriorierrorestimationgoal-orientedadaptivitystochasticGalerkinparametricPDEsrandomcoefficientsmathematicalsoftware
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

T-IFISS is an open-source MATLAB/Octave toolbox whose aim is to make adaptive finite element computation a reproducible laboratory exercise: given an elliptic PDE, a user can run through solve, estimate, mark, and refine with a choice of three a posteriori error estimation strategies and see not just a solution but a controlled error estimate. The paper argues that the same loop extends faithfully to parametric elliptic PDEs with random coefficients, where the spatial mesh and the set of polynomial basis functions in the parameter domain are refined together and the linear algebra is handled by Kronecker-structured preconditioned iteration. Four case studies—an anisotropic diffusion problem, a near-singular harmonic point value, a goal functional on a slit domain, and a random-coefficient diffusion problem—show optimal algebraic convergence rates and bounded effectivity indices when the recommended estimator and marking strategies are used. If the claim is right, the toolbox lets researchers and students compare, validate, and extend adaptive finite element methodology without re-implementing the machinery.

What carries the argument

The machinery that carries the paper is the adaptive loop SOLVE → ESTIMATE → MARK → REFINE, instantiated with hierarchical error estimators (EES1–EES3), edge- or element-based bulk-chasing marking, and newest-vertex bisection with nested finite element spaces. For goal functionals, the dual problem supplies a second Galerkin solution, and the reliable estimate $\mu\zeta$ for $|G(u)-G(u_h)|$ is formed from the product of the primal and dual energy-error estimates; four marking strategies (GO-MARK1–4) combine the primal and dual indicators. For parametric problems, the stochastic Galerkin space is the tensor product $X\otimes P$, the system matrix is a Kronecker sum $A=G_0\otimes K_0+\sum_m G_m\otimes K_m$, and a preconditioned minimum-residual solver works on the component matrices without assembling $A$. The parametric error estimator $\eta$ splits into spatial and parametric contributions, and the adaptive SGFEM algorithm chooses which component to refine based on the dominant contribution or the larger predicted error reduction.

What would settle it

Compute the same four case studies with reference solutions obtained from one additional uniform refinement of the final adaptive mesh (or from a P3/high-order enriched space), and check whether the effectivity indices in Figures 5(b), 9(b), and 12(b) change by more than a few percent. If they shift substantially, the estimators are tracking the reference surrogate rather than the true error.

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Extended reading notes

Core claim

The central discovery is that a single modular software architecture can host the entire adaptive finite element workflow—Galerkin solve, local or global hierarchical error estimation, bulk-chasing marking, and newest-vertex-bisection refinement—and extend it to parametric PDEs through stochastic Galerkin discretization. In the deterministic setting, the paper's key identity is the goal-oriented error estimate $|G(u)-G(u_h)|\le \|u-u_h\|\,\|z-z_h\|$, obtained by Galerkin orthogonality and the dual problem, so that the product of two energy-error estimates gives a rigorous and computable bound on a functional error. In the stochastic setting, the error estimator splits into a spatial part $\|e_X\|_0$ and a parametric part $\|e_P\|_0$, combined as $\eta=(\|e_X\|_0^2+\|e_P\|_0^2)^{1/2}$, and the adaptive algorithm chooses between spatial refinement and enrichment of the polynomial index set by comparing predicted error reductions. The four case studies are presented as evidence that these mechanisms deliver optimal convergence rates and stable effectivity indices in practice.

Load-bearing premise

The reported effectivity indices—the ratios that validate the error estimators—are computed against approximate reference solutions whose own errors are not quantified, so the demonstration of rigorous error control assumes those reference surrogates are accurate enough.

Editorial extensions

If this is right

  • For the anisotropic diffusion problem, the edge-based two-level estimator EES3 with bulk-chasing marking gives an optimal $O(N^{-1/2})$ error-decay rate and essentially monotonic reductions, while the elementwise estimator EES1 stalls because interior residuals vanish for constant coefficients.
  • Goal-oriented adaptivity with the GO-MARK4 combination rule computes the point value $u(0.01,0.01)\approx 1.02679192311$, accurate to nine digits, and keeps the goal-error effectivity index near 2.8.
  • In stochastic Galerkin FEM, refining both the spatial mesh and the parametric index set yields faster convergence than refining only one component; the representative run reached a $3\times 10^{-3}$ tolerance in 19 iterations with 23 polynomials and 6 active parameters.
  • The matrix-free preconditioned solver keeps iteration counts below 20 regardless of mesh resolution and active parameter count, making stochastic Galerkin systems with millions of unknowns practical on a laptop.
  • Because all components are modular and exchangeable, the toolbox supports reproducible research: a user can replace an estimator or marking strategy and immediately compare convergence behavior.

Reading between the lines

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

  • The paper leaves implicit that its effectivity validation rests on unquantified reference surrogates; a direct extension is to recompute Figures 5, 9, and 12 with one additional uniform refinement as the reference and require the effectivity indices to remain stable.
  • The version-2 refinement-selection rule (choose the larger predicted error reduction) could be applied in the goal-oriented stochastic setting, steering the estimator toward a functional by comparing $\mu\zeta$ before and after candidate spatial and parametric refinements.
  • The same Kronecker-preconditioned machinery and hierarchical parametric estimators should carry over to non-affine coefficient representations, giving a path from this toolbox to more general random-field models such as lognormal coefficients.
  • The observed failure of elementwise EES1 on constant-coefficient problems suggests an automatic estimator-selection heuristic: monitor whether interior residuals are structurally zero and switch to an edge-based estimator mid-loop.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces T-IFISS, an open-source MATLAB/Octave toolbox for adaptive finite element computation for deterministic and parametric elliptic PDEs. It describes the modular implementation of the adaptive loop SOLVE–ESTIMATE–MARK–REFINE, three a posteriori error estimation strategies, goal-oriented error estimation with four marking strategies, and stochastic Galerkin FEM with spatial and parametric adaptivity. Four numerical case studies are presented: an anisotropic diffusion problem, a harmonic function in the L-shaped domain, a goal-functional problem on a slit domain, and a parametric diffusion problem with random coefficients. The paper emphasizes that the toolbox is designed both as a research platform for reproducible experiments and as a teaching tool, and it documents the software's directory structure and relationship to the earlier IFISS package.

Significance. The toolbox fills a useful niche by making a range of state-of-the-art adaptive FEM and stochastic Galerkin techniques available in a high-level, readable MATLAB/Octave environment, complementing IFISS and similar packages. The open-source release, modular design, and built-in test problems support reproducibility and classroom use. The numerical case studies provide evidence that the implemented estimators and adaptive strategies achieve optimal or near-optimal convergence rates in the tested examples, and the stochastic Galerkin module demonstrates a genuinely distinctive capability within the IFISS framework. The paper is honest about the distinction between implemented algorithms and their theoretical foundations, which are cited to prior peer-reviewed work.

major comments (2)
  1. [Sections 2.2, 3.3, 4.5; Figures 5(b), 9(b), 12(b)] The effectivity indices are computed against approximate reference solutions rather than exact errors: in Example 1 a P2 adaptive solution with tol=2e-5, in Example 3 a solution on two uniformly refined final meshes, and in Example 4 a P2 solution on the final mesh with the enriched polynomial space PP∪QP (37,020,322 dof). The paper does not quantify the accuracy of these surrogate references. If any of the surrogates is not sufficiently accurate, the reported effectivity indices could be biased, and the empirical support for the phrase 'rigorous error control' would be weakened. Please add a quantitative estimate or bound for the reference error in each case, or explicitly state that the indices are indicative rather than validated surrogates for the exact error.
  2. [Figure 5(b), Figure 12(b)] The effectivity indices for the EES3 estimator in Example 1 are consistently below 1 (roughly 0.5-0.7), and several indices in Example 4 also fall below 1. Since the reference solutions are presumably more accurate than the approximations being tested, these values suggest that the estimator is not a guaranteed upper bound in these examples. The text should clarify whether 'rigorous error control' means a provable upper bound up to a generic constant or merely asymptotic reliability of the estimator, and should state explicitly that the case studies demonstrate the latter if that is the intended interpretation.
minor comments (4)
  1. [Section 4.1, footnote 10] Footnote 10 states that the error estimation module and the adaptive algorithm are 'only implemented for P1 approximation,' but Section 2.2 clearly describes adaptive P2 computation with EES1 using quartic bubble functions in a deterministic setting. Please clarify that this footnote refers to the stochastic Galerkin module, not to the deterministic part of the toolbox.
  2. [Table 3] The multi-indices in Table 3 are displayed as finite tuples without an explicit note that omitted entries are zero. Adding a sentence such as 'for clarity, trailing zero entries are omitted' would make the table easier to read.
  3. [Figure 11 and Section 4.5] The convergence plot in Figure 11 includes a reference line O(N^{-1/3}) for the total degrees of freedom, but the text does not explain the expected rate for the full stochastic Galerkin approximation when both spatial and parametric dimensions contribute to N. A brief comment on how N mixes the two types of degrees of freedom would help interpret the plot.
  4. [General terminology] The paper uses 'effectivity index' for the ratio of the error estimate to an error measured against an approximate reference solution. Since this is not the exact error, consider defining this as a 'surrogate effectivity index' at first use to avoid ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper is a software description whose case studies are independent numerical tests, including an external benchmark; approximate reference solutions create a validation gap, not a circular reduction.

full rationale

The paper does not derive a new result from an input that contains the result. T-IFISS is presented as a toolbox implementing previously published adaptive FEM and stochastic Galerkin algorithms, and its claims are supported by computational case studies rather than by a formal derivation chain. Example 2 is checked against a known exact value from an external NA Digest challenge, which is an independent benchmark. The effectivity studies in Examples 1, 3, and 4 compare the implemented error estimators against higher-fidelity reference solutions (P2 with a tighter tolerance, two uniform refinements, or P2 with an enriched polynomial space); these references are independent of the estimators themselves, so the comparisons are validation experiments rather than self-verification. The authors cite their own prior theoretical work (e.g., [10], [11], [12], [13], [15]) for reliability and convergence properties, but those are published mathematical results with stated assumptions, and the present paper does not rely on them as the conclusion of its own argument. No fitted parameter is renamed as a prediction, and no equation is defined so that the claimed demonstration is forced by construction. The only substantive weakness is that the accuracy of the reference surrogates is not quantified, which could bias reported effectivity indices; however, this is a validation gap and not evidence of circularity.

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

The central claim that the toolbox provides rigorous error control rests on cited theoretical results for the error estimators and marking strategies. The numerical validation also relies on approximate reference solutions whose accuracy is not bounded. No new mathematical theory is introduced, so the ledger lists the domain-specific assumptions inherited from the literature and the surrogate-reliance.

assumptions (4)
  • domain assumption The implemented a posteriori error estimators (EES1-EES3) are reliable and efficient for the tested problems.
    Cited from [1], [6], [43], [44] in Section 2.1; the paper does not re-derive them.
  • domain assumption Goal-oriented marking strategies (GO-MARK2 to GO-MARK4) achieve optimal convergence rates.
    Relies on [28, Theorem 13] cited in Section 3.2.
  • domain assumption Spatial and parametric error estimators in stochastic Galerkin FEM are reliable and provide effectivity.
    Based on [10], [11], [15] as cited in Section 4.3.
  • ad hoc to paper The reference solutions used for effectivity indices are accurate surrogates for the true error.
    Used in Sections 2.2, 3.3, 4.5 (Figures 5b, 9b, 12b); no error bound for the surrogate is given.

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Cite this review

Pith. "Pith review of T-IFISS: a toolbox for adaptive FEM computation." pith.science (2026). https://pith.science/paper/XDCNAPZD

@misc{pith2026190805618,
  author       = {Pith},
  title        = {Pith review of: T-IFISS: a toolbox for adaptive FEM computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDCNAPZD}},
  note         = {Machine review of arXiv:1908.05618}
}
read the original abstract

T-IFISS is a finite element software package for studying finite element solution algorithms for deterministic and parametric elliptic partial differential equations. The emphasis is on self-adaptive algorithms with rigorous error control using a variety of a posteriori error estimation techniques. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration, enabling users to quickly develop new discretizations and test alternative algorithms. The package is also valuable as a teaching tool for students who want to learn about state-of-the-art finite element methodology.

Figures

Figures reproduced from arXiv: 1908.05618 by the authors.

Figure 1
Figure 1. NVB bisections: (a) one, (b)-(c) two, and (d) three bisections of the edges of the triangular [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Example 1: (a) initial coarse mesh; (b) Galerkin solution to problem (2.4). 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Example 1: error estimates at each iteration of the adaptive algorithm employing the error [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Example 1: locally refined meshes produced by the adaptive algorithm employing the error estimation strategies (EES1)–(EES3) and using the element-based D¨orfler marking with θ = 0.5. The header #T` refers to the number of elements in the mesh at step ` of the adaptive…
Figure 5
Figure 5. Figure 5: Example 1: (a) error estimates at each iteration of the adaptive algorithm employing (EES3) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Example 2: (a) Galerkin solution to problem ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Example 3: (a) initial coarse mesh in the GOAFEM algorithm; (b) the primal Galerkin solution. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Example 3: adaptively refined triangulations (top row) and the dual Galerkin solutions (bottom row) computed using the mollifier g0 in (3.5) with r = 0.2, 0.35, 0.5. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Example 3: (a) error estimates at each iteration of the GOAFEM algorithm employing the marking strategy (GO–MARK4); (b) the associated effectivity indices. the GOAFEM algorithm to compute dual Galerkin solutions for different values of the radius r in (3.5). For the SO…
Figure 10
Figure 10. Figure 10: Example 4: (a)–(b) mean and variance of the SGFEM solution; (c) a typical locally refined mesh generated by the adaptive SGFEM algorithm. σ0 = 1). The mean and the variance of an SGFEM solution to this problem are shown in [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Example 4: error estimates at each iteration of the adaptive SGFEM algorithm for different sets of marking parameters. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Example 4: (a) energy error estimates at each iteration, along with their spatial and parametric contributions; (b) the associated effectivity indices [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

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Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    Ainsworth and J

    M. Ainsworth and J. T. Oden , A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley, 2000

    work page 2000

  2. [2]

    Babuˇska, R

    I. Babuˇska, R. Tempone, and E. Zouraris, Galerkin finite element approxima- tions of stochastic elliptic partial differential equations , SIAM J. Numer. Anal., 42 (2004), pp. 800–825

    work page 2004

  3. [3]

    Babuˇska and M

    I. Babuˇska and M. Vogelius , Feedback and adaptive finite element solution of one-dimensional boundary value problems. , Numer. Math., 44 (1984), pp. 75–102

    work page 1984

  4. [4]

    Bangerth, R

    W. Bangerth, R. Hartmann, and G. Kanschat, deal.II – a general purpose ob- ject oriented finite element library, ACM Trans. Math. Software, 33 (2007), pp. 24/1– 24/27. https://www.dealii.org

    work page 2007

  5. [5]

    R. E. Bank , PLTMG: a software package for solving elliptic partial differential equations. Users’ guide 8.0 , vol. 5 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. http: //www.scicomp.ucsd.edu/~reb/software.html

    work page 1998

  6. [6]

    R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985)

    work page 1985

  7. [7]

    B¨ansch, Local mesh refinement in 2 and 3 dimensions , IMPACT Comput

    E. B¨ansch, Local mesh refinement in 2 and 3 dimensions , IMPACT Comput. Sci. Engrg., 3 (1991), pp. 181–191

    work page 1991

  8. [8]

    Becker, E

    R. Becker, E. Estecahandy, and D. Trujillo , Weighted marking for goal- oriented adaptive finite element methods, SIAM J. Numer. Anal., 49 (2011), pp. 2451– 2469

    work page 2011

Show all 58 references
  1. [9]

    Becker and R

    R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods , Acta Numer., 10 (2001), pp. 1–102

    2001

  2. [10]

    Bespalov, C

    A. Bespalov, C. E. Powell, and D. Silvester , Energy norm a posteriori error estimation for parametric operator equations, SIAM J. Sci. Comput., 36 (2014), pp. A339–A363

    2014

  3. [11]

    Bespalov, D

    A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri , Convergence of adaptive stochastic Galerkin FEM, SIAM J. Numer. Anal., 57 (2019), pp. 2359–2382

    2019

  4. [12]

    Bespalov, D

    A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri , Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs, Comput. Methods Appl. Mech. Engrg., 345 (2019), pp. 951–982

    2019

  5. [13]

    Bespalov and L

    A. Bespalov and L. Rocchi, Efficient adaptive algorithms for elliptic PDEs with random data, SIAM/ASA J. Uncertain. Quantif., 6 (2018), pp. 243–272

    2018

  6. [14]

    Available online athttp://web.mat.bham

    , Stochastic T-IFISS, February 2019. Available online athttp://web.mat.bham. ac.uk/A.Bespalov/software/index.html#stoch_tifiss

    2019

  7. [15]

    Bespalov and D

    A. Bespalov and D. Silvester, Efficient adaptive stochastic Galerkin methods for parametric operator equations, SIAM J. Sci. Comput., 38 (2016), pp. A2118–A2140. 23

    2016

  8. [16]

    Bespalov and F

    A. Bespalov and F. Xu , A posteriori error estimation and adaptivity in stochas- tic Galerkin FEM for parametric elliptic PDEs: beyond the affine case . Preprint, arXiv:1903.06520 [math.NA], 2019

    1903 arXiv

  9. [17]

    Blatt, A

    M. Blatt, A. Burchardt, A. Dedner, C. Engwer, J. Fahlke, B. Fle- misch, C. Gersbacher, C. Gr ¨aser, F. Gruber, C. Gr ¨uninger, D. Kempf, R. Kl¨ofkorn, T. Malkmus, S. M ¨uthing, M. Nolte, M. Piatkowski, and O. Sander, The distributed and unified numerics environment, version 2.4...

    2016

  10. [18]

    A. J. Crowder, C. E. Powell, and A. Bespalov, Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation , SIAM J. Sci. Comput., 41 (2019), pp. A1681–A1705

    2019

  11. [19]

    M. K. Deb, I. Babu ˇska, and J. T. Oden , Solution of stochastic partial differ- ential equations using Galerkin finite element techniques , Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6359–6372

    2001

  12. [20]

    D ¨orfler, A convergent adaptive algorithm for Poisson’s equation , SIAM J

    W. D ¨orfler, A convergent adaptive algorithm for Poisson’s equation , SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124

    1996

  13. [21]

    Eigel, C

    M. Eigel, C. J. Gittelson, C. Schwab, and E. Zander , Adaptive stochastic Galerkin FEM, Comput. Methods Appl. Mech. Engrg., 270 (2014), pp. 247–269

    2014

  14. [22]

    , A convergent adaptive stochastic Galerkin finite element method with quasi- optimal spatial meshes , ESAIM Math. Model. Numer. Anal., 49 (2015), pp. 1367– 1398

    2015

  15. [23]

    Eigel and E

    M. Eigel and E. Zander , ALEA – A python framework for spectral methods and low-rank approximations in uncertainty quantification . https://bitbucket.org/ aleadev/alea/src

  16. [24]

    Elman, A

    H. Elman, A. Ramage, and D. Silvester, IFISS: a computational laboratory for investigating incompressible flow problems , SIAM Review, 56 (2014), pp. 261–273

    2014

  17. [25]

    Elman, D

    H. Elman, D. Silvester, and A. Wathen , Finite Elements and Fast Itera- tive Solvers: with Applications in Incompressible Fluid Dynamics , Oxford University Press, Oxford, UK, 2014. Second Edition, xiv+400 pp. ISBN: 978-0-19-967880-8

    2014

  18. [26]

    H. C. Elman, A. Ramage, and D. J. Silvester , Algorithm 886: IFISS, a Matlab toolbox for modelling incompressible flow , ACM Trans. Math. Software, 33 (2007), article 14

    2007

  19. [27]

    Eriksson, D

    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson , Introduction to adaptive methods for differential equations , Acta Numer., 4 (1995), pp. 105–158

    1995

  20. [28]

    Feischl, D

    M. Feischl, D. Praetorius, and K. G. van der Zee , An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal., 54 (2016), pp. 1423–1448

    2016

  21. [29]

    Funken, D

    S. Funken, D. Praetorius, and P. Wissgott , Efficient implementation of adaptive P1-FEM in Matlab , Comput. Methods Appl. Math., 11 (2011), pp. 460–

    2011

  22. [30]

    Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004

    W. Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004

    2004

  23. [31]

    R. G. Ghanem and P. D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag, New York, 1991

    1991

  24. [32]

    M. B. Giles and E. S ¨uli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), pp. 145–236

    2002

  25. [33]

    Gopal and L

    A. Gopal and L. N. Trefethen, New Laplace and Helmholtz solvers , Proc. Nat. Acad. Sci., 116 (2019), pp. 10223–10225

    2019

  26. [34]

    Hecht, New development in FreeFem++, J

    F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), pp. 251–

    2012

  27. [35]

    Holst and S

    M. Holst and S. Pollock , Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems, Numer. Methods Partial Differential Equations, 32 (2016), pp. 479–509

    2016

  28. [36]

    A. Khan, A. Bespalov, C. E. Powell, and D. J. Silvester , Robust a pos- teriori error estimation for stochastic Galerkin formulations of parameter-dependent linear elasticity equations. Preprint, arXiv:1810.07440 [math.NA], 2018

    arXiv

  29. [37]

    A. Khan, C. E. Powell, and D. J. Silvester , Robust preconditioning for stochastic Galerkin formulations of parameter-dependent nearly incompressible elas- ticity equations, SIAM J. Sci. Comput., 41 (2019), pp. A402–A421

    2019

  30. [38]

    Kossaczky, A recursive approach to local mesh refinement in two and three di- mensions, J

    I. Kossaczky, A recursive approach to local mesh refinement in two and three di- mensions, J. Comput. Appl. Math., 55 (1995), pp. 275–288

    1995

  31. [39]

    Logg, K.-A

    A. Logg, K.-A. Mardal, G. N. Wells, et al. , Automated Solution of Differential Equations by the Finite Element Method , Springer, 2012. https: //fenicsproject.org/

    2012

  32. [40]

    G. J. Lord, C. E. Powell, and T. Shardlow, An introduction to computational stochastic PDEs, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2014

    2014

  33. [41]

    W. F. Mitchell, A comparison of adaptive refinement techniques for elliptic prob- lems, ACM Trans. Math. Software, 15 (1989), pp. 326–347

    1989

  34. [42]

    M. S. Mommer and R. Stevenson , A goal-oriented adaptive finite element method with convergence rates, SIAM J. Numer. Anal., 47 (2009), pp. 861–886

    2009

  35. [43]

    Mund and E

    P. Mund and E. P. Stephan , An adaptive two-level method for the coupling of nonlinear FEM-BEM equations, SIAM J. Numer. Anal., 36 (1999), pp. 1001–1021

    1999

  36. [44]

    P. Mund, E. P. Stephan, and J. Weiße, Two-level methods for the single layer potential in R3, Computing, 60 (1998), pp. 243–266

    1998

  37. [45]

    R. H. Nochetto and A. Veeser , Primer of adaptive finite element methods , in Multiscale and Adaptivity: Modeling, Numerics and Applications, vol. 2040, Springer-Verlag Berlin Heidelberg, 2012, pp. 125–225. 25

    2012

  38. [46]

    Persson and G

    P.-O. Persson and G. Strang, A simple mesh generator in Matlab , SIAM Rev., 46 (2004), pp. 329–345. http://persson.berkeley.edu/distmesh/

    2004

  39. [47]

    C. E. Powell and H. C. Elman , Block-diagonal preconditioning for spectral stochastic finite-element systems , IMA Journal of Numerical Analysis, 29 (2009), pp. 350–375

    2009

  40. [48]

    Prudhomme and J

    S. Prudhomme and J. T. Oden , On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors , Comput. Methods Appl. Mech. Engrg., 176 (1999), pp. 313–331. New advances in computational methods (Cachan, 1997)

    1999

  41. [49]

    M. C. Rivara, Mesh refinement processes based on the generalized bisection of sim- plices, SIAM J. Numer. Anal., 21 (1984), pp. 604–613

    1984

  42. [50]

    Rocchi , Adaptive algorithms for partial differential equations with parametric uncertainty, PhD thesis, University of Birmingham, 2019

    L. Rocchi , Adaptive algorithms for partial differential equations with parametric uncertainty, PhD thesis, University of Birmingham, 2019. Electronically published at https://etheses.bham.ac.uk/id/eprint/9157/

    2019

  43. [51]

    Schmidt and K

    A. Schmidt and K. G. Siebert , Design of adaptive finite element software. The finite element toolbox ALBERTA, vol. 42 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2005. http://www.alberta-fem.de

    2005

  44. [52]

    E. G. Sewell, Automatic generation of triangulations for piecewise polynomial ap- proximation, PhD thesis, Purdue University, 1972

    1972

  45. [53]

    D. J. Silvester and V. Simoncini , An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation, ACM Trans. Math. Software, 37 (2011), pp. 42/1–42/22

    2011

  46. [54]

    Stevenson , The completion of locally refined simplicial partitions created by bisection, Math

    R. Stevenson , The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77 (2008), pp. 227–241

    2008

  47. [55]

    L. N. Trefethen, 8-digit Laplace solutions on polygons? Posting on NA Digest at http://www.netlib.org/na-digest-html (29 November 2018)

    2018

  48. [56]

    Zander, SGLib v0.9

    E. Zander, SGLib v0.9. https://github.com/ezander/sglib. 26

  49. [265]

    https://freefem.org/

  50. [490]

    https://www.asc.tuwien.ac.at/~praetorius/matlab/p1afem.zip. 24

Pith tools

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