Abstract
In this work, an r-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of the space-time cylinder, the family of trial spaces that we consider are given as the spans of wavelets-in-time and (locally refined) finite element spaces-in-space. Numerical results illustrate our theoretical findings.
Article PDF
References
Alpert, B. K.: A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24, 246–262 (1993)
Andreev, R.: Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal. 33(1), 242–260 (2013). https://doi.org/10.1093/imanum/drs014
Andreev, R.: Wavelet-in-time multigrid-in-space preconditioning of parabolic evolution equations. SIAM J. Sci. Comput. 38(1), A216–A242 (2016). https://doi.org/10.1137/140998639
Binev, P., DeVore, R.: Fast computation in adaptive tree approximation. Numer. Math. 97(2):193–217 (2004)
Binev, P., Fierro, F., Veeser, A.: Near-best adaptive approximation on conforming meshes. arXiv:1912.13437 (2019)
Beranek, N., Reinhold, M.A., Urban, K.: A space-time variational method for optimal control problems. arXiv:2010.00345 (2020)
Balder, R., Zenger, Ch.: The solution of multidimensional real Helmholtz equations on sparse grids. SIAM. J. Sci. Comput. 17(3), 631–646 (1996)
Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations – Convergence rates. Math. Comp. 70, 27–75 (2001)
Chegini, N. G., Stevenson, R. P.: Adaptive wavelets schemes for parabolic problems: Sparse matrices and numerical results. SIAM J. Numer. Anal. 49(1), 182–212 (2011)
Devaud, D.: Petrov-Galerkin space-time hp-approximation of parabolic equations in H1/2. IMA J. Numer. Anal. 40 (4), 2717–2745 (2020). https://doi.org/10.1093/imanum/drz036
Dyja, R., Ganapathysubramanian, B., van der Zee, K.G.: Parallel-in-space-time, adaptive finite element framework for nonlinear parabolic equations. SIAM J. Sci Comput. 40(3), C283–C304 (2018). https://doi.org/10.1137/16M108985X
Diening, L., Kreuzer, Ch., Stevenson, R.P.: Instance optimality of the adaptive maximum strategy. Found Comput. Math., 1–36. https://doi.org/10.1007/s10208-014-9236-6 (2015)
Dautray, R., Lions, J. -L.: Mathematical analysis and numerical methods for science and technology. Vol. 5. Springer, Berlin. Evolution problems I. https://doi.org/10.1007/978-3-642-58090-1 (1992)
Diening, L., Storn, J.: A space-time DPG method for the heat equation. Comput. Math. Appl., 105:41–53. https://doi.org/10.1016/j.camwa.2021.11.013(2022)
Diening, L., Storn, J., Tscherpel, T.: On the Sobolev and Lp-stability of the L2-projection. SIAM J. Numer. Anal. 59 (5), 2571–2607 (2021). https://doi.org/10.1137/20M1358013
Dahmen, W., Stevenson, R. P., Westerdiep, J.: Accuracy controlled data assimilation for parabolic problems (2021) 40 pages Accepted for publication in. Math. Comp. arXiv:2105.05836
Ern, A., Smears, I., Vohralík, M.: Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems. SIAM J. Numer. Anal. 55(6), 2811–2834 (2017). https://doi.org/10.1137/16M1097626
Führer, T., Karkulik, M.: Space-time least-squares finite elements for parabolic equations. Comput. Math. Appl. 92, 27–36 (2021). https://doi.org/10.1016/j.camwa.2021.03.004
Gunzburger, M. D., Kunoth, A.: Space-time adaptive wavelet methods for control problems constrained by parabolic evolution equations. SIAM J. Contr. Optim. 49(3), 1150–1170 (2011)
Griebel, M., Oeltz, D.: A sparse grid space-time discretization scheme for parabolic problems. Computing 81(1), 1–34 (2007)
Gimperlein, H., Stocek, J.: Space-time adaptive finite elements for nonlocal parabolic variational inequalities. Comput. Methods Appl. Mech Engrg. 352, 137–171 (2019). https://doi.org/10.1016/j.cma.2019.04.019
Hofer, Ch., Langer, U., Neumüller, M., Schneckenleitner, R.: Parallel and robust preconditioning for space-time isogeometric analysis of parabolic evolution problems. SIAM J. Sci. Comput. 41(3), A1793–A1821 (2019). https://doi.org/10.1137/18M1208794
Kondratyuk, Y., Stevenson, R.P.: An optimal adaptive finite element method for the Stokes problem. SIAM J. Numer. Anal. 46(2), 747–775 (2008)
Kestler, S., Stevenson, R.P.: Fast evaluation of system matrices w.r.t. multi-tree collections of tensor product refinable basis functions. J. Comput. Appl. Math. 260, 103–116 (2014)
Kestler, S., Steih, K., Urban, K.: An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations. Math. Comp. 85(299), 1309–1333 (2016). https://doi.org/10.1090/mcom/3009
Larsson, S., Molteni, M.: Numerical solution of parabolic problems based on a weak space-time formulation. Comput. Methods Appl. Math. 17(1), 65–84 (2017). https://doi.org/10.1515/cmam-2016-0027
Langer, U., Moore, S. E., Neumüller, M.: Space-time isogeometric analysis of parabolic evolution problems. Comput. Methods Appl. Mech. Engrg. 306, 342–363 (2016). https://doi.org/10.1016/j.cma.2016.03.042
Langer, U., Schafelner, A.: An optimal adaptive finite element method for Non-autonomous Parabolic Problems with Distributional Sources. Comput. Methods Appl. Math. 20(4), 677–693 (2020). https://doi.org/10.1515/cmam-2020-0042
Neumüller, M., Smears, I.: Time-parallel iterative solvers for parabolic evolution equations. SIAM J. Sci. Comput. 41(1), C28–C51 (2019). https://doi.org/10.1137/18M1172466
Olshanskii, M. A., Reusken, A.: On the convergence of a multigrid method for linear reaction-diffusion problems. Computing 65(3), 193–202 (2000). https://doi.org/10.1007/s006070070006
Pearson, J. W., Wathen, A. J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 19(5), 816–829 (2012). https://doi.org/10.1002/nla.814
Rekatsinas, N., Stevenson, R.P.: An optimal adaptive tensor product wavelet solver of a space-time FOSLS formulation of parabolic evolution problems. Adv. Comput. Math. 45(2), 1031–1066 (2019). https://doi.org/10.1007/s10444-018-9644-2
Schwab, Ch., Stevenson, R.P.: A space-time adaptive wavelet method for parabolic evolution problems. Math Comp. 78, 1293–1318 (2009). https://doi.org/10.1090/S0025-5718-08-02205-9
Stevenson, R. P.: The frequency decomposition multi-level method: a robust additive hierarchical basis preconditioner. Math. Comp. 65(215), 983–997 (1996)
Stevenson, R. P.: The completion of locally refined simplicial partitions created by bisection. Math. Comp. 77, 227–241 (2008)
Steinbach, O.: Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15(4), 551–566 (2015). https://doi.org/10.1515/cmam-2015-0026
Stevenson, R. P., van Venetië, R.: Uniform preconditioners for problems of negative order. Math. Comp. 89(322), 645–674 (2020). https://doi.org/10.1090/mcom/3481
Stevenson, R. P., Westerdiep, J.: Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations. IMA J. Numer. Anal. 41(1), 28–47 (2021). https://doi.org/10.1093/imanum/drz069
Steinbach, O., Yang, H.: Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D. Numer. Linear Algebra Appl. 25(3), e2143, 17 (2018). https://doi.org/10.1002/nla.2143
Steinbach, O., Zank, M.: Coercive space-time finite element methods for initial boundary value problems. Electron. Trans. Numer. Anal. 52:154–194. https://doi.org/10.1553/etna_vol52s154 (2020)
van Venetië, R., Westerdiep, J.: A parallel algorithm for solving linear parabolic evolution equations. 33–50, Springer Proc. Math. Stat., 356, Springer, Cham (2021)
van Venetië, R., Westerdiep, J.: Efficient space-time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space. arXiv:2104.08143 (2021)
Wloka, J.: Partielle Differentialgleichungen. B. G. Teubner, Stuttgart. Sobolevräume und Randwertaufgaben (1982)
Wu, J., Zheng, H.: Uniform convergence of multigrid methods for adaptive meshes. Appl. Numer Math. 113, 109–123 (2017). https://doi.org/10.1016/j.apnum.2016.11.005
Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D., Calo, V. M.: A class of discontinuous Petrov-Galerkin methods. Part IV: the optimal test norm and time-harmonic wave propagation in 1D. J. Comput. Phys. 230(7), 2406–2432 (2011)
Funding
The second and third authors have been supported by the Netherlands Organization for Scientific Research (NWO) under contract. no. 613.001.652
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Communicated by: Peter Benner
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Stevenson, R., van Venetië, R. & Westerdiep, J. A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations. Adv Comput Math 48, 17 (2022). https://doi.org/10.1007/s10444-022-09930-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-022-09930-w
Keywords
- Space-time variational formulations of parabolic PDEs
- Quasi-best approximations
- Least squares methods
- Adaptive approximation
- Tensor product approximation
- Optimal preconditioners