×

zbMATH — the first resource for mathematics

Extended group finite element method. (English) Zbl 07311175
Summary: Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many cases, this allows for the exact reformulation of the discrete nonlinear problem into a quadratic problem with algebraic constraints. Furthermore, the substitution of the nonlinear terms often shifts general nonlinear forms into trilinear forms, which can easily be described by third-order tensors. The numerical benefits as well as the advantages in comparison to the original group finite element method are studied using a wide variety of academic benchmark problems.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J60 Nonlinear elliptic equations
65D05 Numerical interpolation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albersmeyer, J.; Diehl, M., The lifted Newton method and its application in optimization, SIAM J. Optim., 20, 1655-1684 (2010) · Zbl 1198.90396
[2] Antil, H.; Heinkenschloss, M.; Sorensen, D. C., Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems, 101-136 (2014), Springer International Publishing · Zbl 1312.65180
[3] Bader, B. W.; Kolda, T. G., Efficient MATLAB computations with sparse and factored tensors, SIAM J. Sci. Comput., 30, 205-231 (2007) · Zbl 1159.65323
[4] Bader, B. W.; Kolda, T. G., MATLAB tensor toolbox version 3.1 (June 2019), Available online
[5] Barrenechea, G. R.; Knobloch, P., Analysis of a group finite element formulation, Appl. Numer. Math., 118, 238-248 (2017) · Zbl 1367.65141
[6] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics (2008), Springer: Springer New York · Zbl 1135.65042
[7] Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 2737-2764 (2010) · Zbl 1217.65169
[8] Christie, I.; Griffiths, D. F.; Mitchell, A. R.; Sanz-Serna, J. M., Product approximation for non-linear problems in the finite element method, IMA J. Numer. Anal., 1, 253-266 (1981) · Zbl 0469.65072
[9] Dickinson, B.; Singler, J., Nonlinear model reduction using group proper orthogonal decomposition, Int. J. Numer. Anal. Model., 7, 356-372 (2010)
[10] Douglas, J.; Dupont, T., The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures, Math. Comput., 29, 360-389 (1975) · Zbl 0311.65060
[11] Dunavant, D. A., High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int. J. Numer. Methods Eng., 21, 1129-1148 (1985) · Zbl 0589.65021
[12] Ern, A.; Guermond, J., Theory and Practice of Finite Elements, Applied Mathematical Sciences (2004), Springer: Springer New York
[13] Evans, L., Partial Differential Equations, Graduate Studies in Mathematics (2010), American Mathematical Society
[14] Farina, A., Chapter 2 - Liouville-type theorems for elliptic problems, (Chipot, M., Stationary Partial Differential Equations. Stationary Partial Differential Equations, Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4 (2007), North-Holland), 61-116 · Zbl 1191.35128
[15] Fletcher, C. A.J., The group finite element formulation, Comput. Methods Appl. Mech. Eng., 37, 225-244 (1983)
[16] Geuzaine, C.; Remacle, J.-F., Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 1309-1331 (2009) · Zbl 1176.74181
[17] Gu, C., QLMOR: a projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 30, 1307-1320 (2011)
[18] Hernández, J. A.; Caicedo, M. A.; Ferrer, A., Dimensional hyper-reduction of nonlinear finite element models via empirical cubature, Comput. Methods Appl. Mech. Eng., 313, 687-722 (2017) · Zbl 1439.74419
[19] Hintermüller, M.; Vicente, L. N., Space mapping for optimal control of partial differential equations, SIAM J. Optim., 15, 1002-1025 (2005) · Zbl 1114.49033
[20] Houston, P.; Wihler, T. P., Discontinuous Galerkin methods for problems with Dirac delta source, ESAIM: M2AN, 46, 1467-1483 (2012) · Zbl 1272.65092
[21] Ito, K.; Kunisch, K., Augmented Lagrangian-SQP methods for nonlinear optimal control problems of tracking type, SIAM J. Control Optim., 34, 874-891 (1996) · Zbl 0860.49023
[22] Kawohl, B., On a family of torsional creep problem, J. Reine Angew. Math., 1990, 1-22 (1990) · Zbl 0701.35015
[23] Kramer, B.; Willcox, K. E., Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA J., 57, 2297-2307 (2019)
[24] Kuzmin, D., Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes, J. Comput. Appl. Math., 236, 2317-2337 (2012) · Zbl 1241.65083
[25] Logg, A.; Mardal, K. A.; Wells, G., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lecture Notes in Computational Science and Engineering (2012), Springer Berlin Heidelberg · Zbl 1247.65105
[26] Roache, P. J., The method of manufactured solutions for code verification, (Beisbart, C.; Saam, N. J., Computer Simulation Validation: Fundamental Concepts, Methodological Frameworks, and Philosophical Perspectives (2019), Springer International Publishing), 295-318
[27] Roubíček, T., Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics (2013), Springer: Springer Basel · Zbl 1270.35005
[28] Scott, R., Finite element convergence for singular data, Numer. Math., 21, 317-327 (1973) · Zbl 0255.65037
[29] Tiso, P.; Dedden, R.; Rixen, D., A modified discrete empirical interpolation method for reducing non-linear structural finite element models, (Volume 7B: 9th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2013))
[30] Tolle, K.; Marheineke, N., On online parameter identification in laser-induced thermotherapy, (Pinnau, R.; Klar, A.; Gauger, N. R., Modeling, Simulation and Optimization in the Health- and Energy-Sector (2021), Springer)
[31] Wang, Z., Nonlinear model reduction based on the finite element method with interpolated coefficients: semilinear parabolic equations, Numer. Methods Partial Differ. Equ., 31, 1713-1741 (2015) · Zbl 1334.65159
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.