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T-IFISS: a toolbox for adaptive FEM computation. (English) Zbl 07288719
Summary: 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.

MSC:
65 Numerical analysis
74 Mechanics of deformable solids
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[1] Elman, H. C.; Ramage, A.; Silvester, D. J., Algorithm 886: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33, 2, Article 14 pp. (2007) · Zbl 1365.65326
[2] Elman, H.; Ramage, A.; Silvester, D., IFISS: a computational laboratory for investigating incompressible flow problems, SIAM Rev., 56, 2, 261-273 (2014) · Zbl 1426.76645
[3] Elman, H.; Silvester, D.; Wathen, A., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, xiv+400 (2014), Oxford University Press: Oxford University Press Oxford, UK · Zbl 1304.76002
[4] Bangerth, W.; Hartmann, R.; Kanschat, G., Deal.II - a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33, 4, 24/1-24/27 (2007), https://www.dealii.org · Zbl 1365.65248
[5] Blatt, M.; Burchardt, A.; Dedner, A.; Engwer, C.; Fahlke, J.; Flemisch, B.; Gersbacher, C.; Gräser, C.; Gruber, F.; Grüninger, C.; Kempf, D.; Klöfkorn, R.; Malkmus, T.; Müthing, S.; Nolte, M.; Piatkowski, M.; Sander, O., The distributed and unified numerics environment, version 2.4, Arch. Num. Soft., 4, 100, 13-29 (2016), https://www.dune-project.org/
[6] Logg, A.; Mardal, K.-A.; Wells, G. N., Automated Solution of Differential Equations by the Finite Element Method (2012), Springer, https://fenicsproject.org/
[7] Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 3-4, 251-265 (2012), https://freefem.org/ · Zbl 1266.68090
[8] Schmidt, A.; Siebert, K. G., (Design of Adaptive Finite Element Software. the Finite Element Toolbox ALBERTA. Design of Adaptive Finite Element Software. the Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, vol. 42 (2005), Springer-Verlag: Springer-Verlag Berlin), xii+315, http://www.alberta-fem.de · Zbl 1068.65138
[9] Bank, R. E., (PLTMG: a software package for solving elliptic partial differential equations. Users’ guide 8.0. PLTMG: a software package for solving elliptic partial differential equations. Users’ guide 8.0, Software, Environments, and Tools, vol. 5 (1998), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA), xii+110, http://www.scicomp.ucsd.edu/ reb/software.html · Zbl 0990.65500
[10] Funken, S.; Praetorius, D.; Wissgott, P., Efficient implementation of adaptive P1-FEM in Matlab, Comput. Methods Appl. Math., 11, 4, 460-490 (2011), https://www.asc.tuwien.ac.at/ praetorius/matlab/p1afem.zip · Zbl 1284.65197
[11] Bespalov, A.; Rocchi, L., Stochastic T-IFISS (2019), Available online at http://web.mat.bham.ac.uk/A.Bespalov/software/index.html#stoch_tifiss
[12] M. Eigel, E. Zander, ALEA - A python framework for spectral methods and low-rank approximations in uncertainty quantification, https://bitbucket.org/aleadev/alea/src.
[13] E. Zander, SGLib v0.9, https://github.com/ezander/sglib.
[14] Rocchi, L., Adaptive algorithms for partial differential equations with parametric uncertainty (2019), University of Birmingham, Electronically published at https://etheses.bham.ac.uk/id/eprint/9157/
[15] Persson, P.-O.; Strang, G., A simple mesh generator in Matlab, SIAM Rev., 46, 2, 329-345 (2004), http://persson.berkeley.edu/distmesh/ · Zbl 1061.65134
[16] Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics (New York) (2000), Wiley · Zbl 1008.65076
[17] Bank, R. E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, 170 (1985) · Zbl 0569.65079
[18] Mund, P.; Stephan, E. P.; Weiße, J., Two-level methods for the single layer potential in \(\mathbb{R}^3\), Computing, 60, 3, 243-266 (1998) · Zbl 0901.65072
[19] Mund, P.; Stephan, E. P., An adaptive two-level method for the coupling of nonlinear FEM-BEM equations, SIAM J. Numer. Anal., 36, 4, 1001-1021 (1999) · Zbl 0938.65138
[20] Babuška, I.; Vogelius, M., Feedback and adaptive finite element solution of one-dimensional boundary value problems., Numer. Math., 44, 75-102 (1984) · Zbl 0574.65098
[21] Dörfler, W., A convergent adaptive algorithm for poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124 (1996) · Zbl 0854.65090
[22] Sewell, E. G., Automatic Generation of Triangulations for Piecewise Polynomial Approximation (1972), Purdue University, (Ph.D. thesis)
[23] Rivara, M. C., Mesh refinement processes based on the generalized bisection of simplices, SIAM J. Numer. Anal., 21, 3, 604-613 (1984) · Zbl 0574.65133
[24] Bänsch, E., Local mesh refinement in 2 and 3 dimensions, IMPACT Comput. Sci. Engrg., 3, 3, 181-191 (1991) · Zbl 0744.65074
[25] Kossaczky, I., A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math., 55, 275-288 (1995) · Zbl 0823.65119
[26] Stevenson, R., The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77, 261, 227-241 (2008) · Zbl 1131.65095
[27] Nochetto, R. H.; Veeser, A., Primer of adaptive finite element methods, (Multiscale and Adaptivity: Modeling, Numerics and Applications, Vol. 2040 (2012), Springer-Verlag Berlin Heidelberg), 125-225 · Zbl 1252.65192
[28] Mitchell, W. F., A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Softw., 15, 4, 326-347 (1989) · Zbl 0900.65306
[29] Trefethen, L. N., 8-digit Laplace solutions on polygons? (2018), Posting on NA Digest at http://www.netlib.org/na-digest-html (29 November 2018)
[30] Gopal, A.; Trefethen, L. N., New Laplace and Helmholtz solvers, Proc. Natl. Acad. Sci., 116, 21, 10223-10225 (2019) · Zbl 1431.65224
[31] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta Numer., 4, 105-158 (1995) · Zbl 0829.65122
[32] Prudhomme, S.; Oden, J. T., On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg., 176, 1-4, 313-331 (1999), New advances in computational methods (Cachan, 1997) · Zbl 0945.65123
[33] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10, 1-102 (2001) · Zbl 1105.65349
[34] Giles, M. B.; Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11, 145-236 (2002) · Zbl 1105.65350
[35] Holst, M.; Pollock, S., Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems, Numer. Methods Partial Differential Equations, 32, 2, 479-509 (2016) · Zbl 1337.65139
[36] Mommer, M. S.; Stevenson, R., A goal-oriented adaptive finite element method with convergence rates, SIAM J. Numer. Anal., 47, 2, 861-886 (2009) · Zbl 1195.65174
[37] Becker, R.; Estecahandy, E.; Trujillo, D., Weighted marking for goal-oriented adaptive finite element methods, SIAM J. Numer. Anal., 49, 6, 2451-2469 (2011) · Zbl 1245.65155
[38] Feischl, M.; Praetorius, D.; van der Zee, K. G., An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal., 54, 3, 1423-1448 (2016) · Zbl 1382.65392
[39] Ghanem, R. G.; Spanos, P. D., Stochastic Finite Elements: A Spectral Approach (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0722.73080
[40] Deb, M. K.; Babuška, I.; Oden, J. T., Solution of stochastic partial differential equations using galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg., 190, 6359-6372 (2001) · Zbl 1075.65006
[41] Babuška, I.; Tempone, R.; Zouraris, E., Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42, 2, 800-825 (2004) · Zbl 1080.65003
[42] Lord, G. J.; Powell, C. E.; Shardlow, T., (An Introduction to Computational Stochastic PDEs. An Introduction to Computational Stochastic PDEs, Cambridge Texts in Applied Mathematics (2014), Cambridge University Press: Cambridge University Press New York), xii+503 · Zbl 1327.60011
[43] Eigel, M.; Gittelson, C. J.; Schwab, C.; Zander, E., Adaptive stochastic galerkin FEM, Comput. Methods Appl. Mech. Engrg., 270, 247-269 (2014) · Zbl 1296.65157
[44] Gautschi, W., (Orthogonal Polynomials: Computation and Approximation. Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation (2004), Oxford University Press: Oxford University Press New York) · Zbl 1130.42300
[45] Silvester, D. J.; Simoncini, V., An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation, ACM Trans. Math. Softw., 37, 4, 42/1-42/22 (2011) · Zbl 1365.65085
[46] Powell, C. E.; Elman, H. C., Block-diagonal preconditioning for spectral stochastic finite-element systems, IMA J. Numer. Anal., 29, 2, 350-375 (2009) · Zbl 1169.65007
[47] Eigel, M.; Gittelson, C. J.; Schwab, C.; Zander, E., A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes, ESAIM Math. Model. Numer. Anal., 49, 5, 1367-1398 (2015) · Zbl 1335.65006
[48] Bespalov, A.; Powell, C. E.; Silvester, D., Energy norm a posteriori error estimation for parametric operator equations, SIAM J. Sci. Comput., 36, 2, A339-A363 (2014)
[49] Bespalov, A.; Silvester, D., Efficient adaptive stochastic galerkin methods for parametric operator equations, SIAM J. Sci. Comput., 38, 4, A2118-A2140 (2016) · Zbl 1416.65435
[50] Bespalov, A.; Praetorius, D.; Rocchi, L.; Ruggeri, M., Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs, Comput. Methods Appl. Mech. Engrg., 345, 951-982 (2019) · Zbl 1440.65184
[51] Bespalov, A.; Rocchi, L., Efficient adaptive algorithms for elliptic PDEs with random data, SIAM/ASA J. Uncertain. Quantif., 6, 1, 243-272 (2018) · Zbl 1398.65289
[52] Bespalov, A.; Praetorius, D.; Rocchi, L.; Ruggeri, M., Convergence of adaptive stochastic galerkin FEM, SIAM J. Numer. Anal., 57, 5, 2359-2382 (2019) · Zbl 1425.65146
[53] Khan, A.; Powell, C. E.; Silvester, D. J., Robust preconditioning for stochastic galerkin formulations of parameter-dependent nearly incompressible elasticity equations, SIAM J. Sci. Comput., 41, 1, A402-A421 (2019) · Zbl 1428.65087
[54] Khan, A.; Bespalov, A.; Powell, C. E.; Silvester, D. J., Robust a posteriori error estimation for stochastic galerkin formulations of parameter-dependent linear elasticity equations (2018), Preprint, arXiv:1810.07440 [math.NA]
[55] Bespalov, A.; Xu, F., A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case (2019), Preprint, arXiv:1903.06520 [math.NA]
[56] Crowder, A. J.; Powell, C. E.; Bespalov, A., Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation, SIAM J. Sci. Comput., 41, A1681-A1705 (2019) · Zbl 1420.65120
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