An optimal adaptive tensor product wavelet solver of a space-time FOSLS formulation of parabolic evolution problems

  • Nikolaos Rekatsinas
  • Rob Stevenson
Open Access


In this work, we construct a well-posed first-order system least squares (FOSLS) simultaneously space-time formulation of parabolic PDEs. Using an adaptive wavelet solver, this problem is solved with the best possible rate in linear complexity. Thanks to the use of a basis that consists of tensor products of wavelets in space and time, this rate is equal to that when solving the corresponding stationary problem. Our findings are illustrated by numerical results.


Parabolic PDEs Space-time variational formulation First order system least squares Adaptive wavelet solver Optimal rates Linear complexity 

Mathematics Subject Classification (2010)

35K20 41A25 41A63 42C40 65N12 65T60 65N30 


  1. 1.
    Aimar, H., Gómez, I.: Parabolic Besov regularity for the heat equation. Constr. Approx. 36(1), 145–159 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alpert, B.: A class of bases in L 2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24, 246–262 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andreev, R.: Space-time discretization of the heat equation. Numer. Algorithms 67(4), 713–731 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Binev, P., DeVore, R.: Fast computation in adaptive tree approximation. Numer. Math. 97(2), 193–217 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Babuška, I., Janik, T.: The h-p version of the finite element method for parabolic equations. I. The p-version in time. Numer. Methods Partial Differential Equations 5(4), 363–399 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Babuška, I., Janik, T.: The h-p version of the finite element method for parabolic equations. II. The h-p version in time. Numer. Methods Partial Differential Equations 6(4), 343–369 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Balder, R., Zenger, C.: The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput. 17(3), 631–646 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations – convergence rates, vol. 70 (2001)Google Scholar
  10. 10.
    Cioica, P., Dahlke, S., Döhring, N., Friedrich, U., Kinzel, S., Lindner, F., Raasch, T., Ritter, K., Schilling, R.: Convergence analysis of spatially adaptive Rothe methods. Found. Comput. Math. 14(5), 863–912 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cohen, A., Dahmen, W., Daubechies, I., DeVore, R.: Tree approximation and optimal encoding. Appl. Comput. Harmon. Anal. 11(2), 192–226 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chegini, N., Dahlke, S., Friedrich, U., Stevenson, R.: Piecewise tensor product wavelet bases by extensions and approximation rates. Math. Comp. 82, 2157–2190 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chegini, N., Stevenson, R.: Adaptive wavelets schemes for parabolic problems: sparse matrices and numerical results. SIAM J. Numer. Anal. 49(1), 182–212 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chegini, N., Stevenson, R.: An adaptive wavelet method for semi-linear first-order system least squares. Comput. Methods Appl. Math. 15(4), 439–463 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dörfler, W., Findeisen, S., Wieners, C.: Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Comput. Methods Appl. Math. 16(3), 409–428 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer, Berlin (1992). Evolution problems IzbMATHGoogle Scholar
  17. 17.
    Dahmen, W., Stevenson, R.: Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal. 37(1), 319–352 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dauge, M., Stevenson, R.: Sparse tensor product wavelet approximation of singular functions. SIAM J. Math. Anal. 42(5), 2203–2228 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ellis, T., Chan, J., Demkowicz, L.: Robust DPG methods for transient convection-diffusion, in Building bridges: connections and challenges in modern approaches to numerical partial differential equations. Lect. Notes Comput. Sci. Eng. 114, 179–203 (2016). SpringerCrossRefGoogle Scholar
  20. 20.
    Gantumur, T., Harbrecht, H., Stevenson, R.: An optimal adaptive wavelet method without coarsening of the iterands. Math. Comp. 76, 615–629 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gunzburger, M., Kunoth, A.: Space-time adaptive wavelet methods for control problems constrained by parabolic evolution equations. J. Contr. Optim. 49(3), 1150–1170 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gander, M., Neumüller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38(4), A2173–A2208 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Griebel, M., Oswald, P.: Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4(1–2), 171–206 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Griebel, M., Oeltz, D.: A sparse grid space-time discretization scheme for parabolic problems. Computing 81(1), 1–34 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kestler, S., Stevenson, R.: 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)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kestler, S., Steih, K., Urban, K.: An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations. Math. Comput. 85(299), 1309–1333 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Langer, U., Moore, S., Neumüller, M.: Space-time isogeometric analysis of parabolic evolution problems. Comput. Methods Appl. Mech. Engrg. 306, 342–363 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Majidi, M., Starke, G.: Least-squares Galerkin methods for parabolic problems. II. The fully discrete case and adaptive algorithms. SIAM J. Numer. Anal. 39(5), 1648–1666 (2001/02)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Majidi, M., Starke, G.: Least-squares Galerkin methods for parabolic problems. I. Semidiscretization in time. SIAM J. Numer. Anal. 39(4), 1302–1323 (2001)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Messner, M., Schanz, M., Tausch, J.: A fast Galerkin method for parabolic space-time boundary integral equations. J. Comput. Phys. 258, 15–30 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nitsche, P.-A.: Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx. 24(1), 49–70 (2006)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Nguyen, H., Stevenson, R.: Finite element wavelets with improved quantitative properties. J. Comput. Appl. Math. 230(2), 706–727 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Rekatsinas, N.: Optimal adaptive wavelet methods for solving first order system least squares, PhD thesis University of Amsterdam (2018)Google Scholar
  34. 34.
    Rekatsinas, N., Stevenson, R.: An optimal adaptive wavelet method for first order system least squares. Numer. Math. 140(1), 191–237 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Rekatsinas, N., Stevenson, R.: A quadratic finite element wavelet Riesz basis. Int. J. Wavelets Multiresolution Inf. Process. 16(4), 1850033, 17 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schwab, C., Stevenson, R.: A space-time adaptive wavelet method for parabolic evolution problems. Math. Comp. 78, 1293–1318 (2009)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Schwab, C., Stevenson, R.: Fractional space-time variational formulations of (Navier)-Stokes equations. SIAM J. Math. Anal. 49(4), 2442–2467 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Stevenson, R.: Stable three-point wavelet bases on general meshes. Numer. Math. 80, 131–158 (1998)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Stevenson, R.: Adaptive wavelet methods for linear and nonlinear least-squares problems, Found. Comput. Math. 14(2), 237–283 (2014)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Steinbach, O.: Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15(4), 551–566 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Sickel, W., Ullrich, T.: Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory 161, 748–786 (2009)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Volume 68 of Applied Mathematical Sciences, 2nd edn. Springer, New York (1997)CrossRefGoogle Scholar

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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