Convergence of Adaptive Stochastic Galerkin FEM
Abstract
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.
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References
1.
M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math., Wiley, New York, 2000, https://doi.org/10.1002/9781118032824.
2.
I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. Math., 44 (1984), pp. 75--102, https://doi.org/10.1007/BF01389757.
3.
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, https://doi.org/10.1137/130916849.
4.
A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri, Convergence of Adaptive Stochastic Galerkin FEM, 2018, https://arxiv.org/abs/1811.09462.
5.
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, https://doi.org/10.1016/j.cma.2018.10.041.
6.
A. Bespalov and L. Rocchi, Efficient adaptive algorithms for elliptic PDEs with random data, SIAM/ASA J. Uncertain. Quantif., 6 (2018), pp. 243--272, https://doi.org/10.1137/17M1139928.
7.
A. Bespalov and L. Rocchi, Stochastic T-IFISS, http://web.mat.bham.ac.uk/A.Bespalov/software/index.html#stoch_tifiss (2019).
8.
A. Bespalov and D. Silvester, Efficient adaptive stochastic Galerkin methods for parametric operator equations, SIAM J. Sci. Comput., 38 (2016), pp. A2118--A2140, https://doi.org/10.1137/15M1027048.
9.
T. Butler, P. Constantine, and T. Wildey, A posteriori error analysis of parameterized linear systems using spectral methods, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 195--209, https://doi.org/10.1137/110840522.
10.
J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524--2550, https://doi.org/10.1137/07069047X.
11.
A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer., 24 (2015), pp. 1--159, https://doi.org/10.1017/S0962492915000033.
12.
A. Cohen, R. DeVore, and C. Schwab, Convergence rates of best $N$-term Galerkin approximations for a class of elliptic sPDEs, Found. Comput. Math., 10 (2010), pp. 615--646, https://doi.org/10.1007/s10208-010-9072-2.
13.
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, https://doi.org/10.1137/18M1194420.
14.
W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106--1124, https://doi.org/10.1137/0733054.
15.
M. Eigel, C. J. Gittelson, C. Schwab, and E. Zander, Adaptive stochastic Galerkin FEM, Comput. Methods Appl. Mech. Engrg., 270 (2014), pp. 247--269, https://doi.org/10.1016/j.cma.2013.11.015.
16.
M. Eigel, C. J. Gittelson, C. Schwab, and E. Zander, A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes, ESAIM Math. Model. Numer. Anal., 49 (2015), pp. 1367--1398, https://doi.org/10.1051/m2an/2015017.
17.
M. Eigel and C. Merdon, Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin finite element methods, SIAM/ASA J. Uncertain. Quantif., 4 (2016), pp. 1372--1397, https://doi.org/10.1137/15M102188X.
18.
C. Erath, G. Gantner, and D. Praetorius, Optimal convergence behavior of adaptive FEM driven by simple $(h-h/2)$-type error estimators, Comput. Math. Appl., (2019), https://doi.org/10.1016/j.camwa.2019.07.014.
19.
D. Estep, M. J. Holst, and A. M\aalqvist, Nonparametric density estimation for randomly perturbed elliptic problems III: Convergence, computational cost, and generalizations, J. Appl. Math. Comput., 38 (2012), pp. 367--387, https://doi.org/10.1007/s12190-011-0484-1.
20.
D. Estep, A. M\aalqvist, and S. Tavener, Nonparametric density estimation for randomly perturbed elliptic problems I: Computational methods, a posteriori analysis, and adaptive error control, SIAM J. Sci. Comput., 31 (2009), pp. 2935--2959, https://doi.org/10.1137/080731670.
21.
D. Estep, A. M\aalqvist, and S. Tavener, Nonparametric density estimation for randomly perturbed elliptic problems II: Applications and adaptive modeling, Internat. J. Numer. Methods Engrg., 80 (2009), pp. 846--867, https://doi.org/10.1002/nme.2547.
22.
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, https://doi.org/10.1137/15M1021982.
23.
M. B. Giles and E. Süli, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), pp. 145--236, https://doi.org/10.1017/CBO9780511550140.003.
24.
C. J. Gittelson, An adaptive stochastic Galerkin method for random elliptic operators, Math. Comp., 82 (2013), pp. 1515--1541, https://doi.org/10.1090/S0025-5718-2013-02654-3.
25.
M. Karkulik, D. Pavlicek, and D. Praetorius, On 2D newest vertex bisection: Optimality of mesh-closure and $H^1$-stability of $L_2$-projection, Constr. Approx., 38 (2013), pp. 213--234, https://doi.org/10.1007/s00365-013-9192-4.
26.
A. Khan, A. Bespalov, C. E. Powell, and D. J. Silvester, Robust A Posteriori Error Estimation for Stochastic Galerkin Formulations of Parameter-Dependent Linear Elasticity Equations, 2018, https://arxiv.org/abs/1810.07440.
27.
L. Mathelin and O. Le Ma\^\itre, Dual-based a posteriori error estimate for stochastic finite element methods, Commun. Appl. Math. Comput. Sci., 2 (2007), pp. 83--115, https://doi.org/10.2140/camcos.2007.2.83.
28.
P. Morin, R. H. Nochetto, and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466--488, https://doi.org/10.1137/S0036142999360044.
29.
P. Morin, K. G. Siebert, and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 18 (2008), pp. 707--737, https://doi.org/10.1142/S0218202508002838.
30.
C. Schwab and C. J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numer., 20 (2011), pp. 291--467, https://doi.org/10.1017/S0962492911000055.
31.
R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77 (2008), pp. 227--241, https://doi.org/10.1090/S0025-5718-07-01959-X.
32.
X. Wan and G. E. Karniadakis, Error control in multi-element generalized polynomial chaos method for elliptic problems with random coefficients, Commun. Comput. Phys., 5 (2009), pp. 793--820.
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© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 4 December 2018
Accepted: 25 July 2019
Published online: 3 October 2019
Keywords
MSC codes
Authors
Funding Information
Austrian Science Fund https://doi.org/10.13039/501100002428 : W1245, F65
Funding Information
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/P013791/1
Funding Information
Alan Turing Institute https://doi.org/10.13039/100012338 : EP/N510129/1
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