An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations

Kestler, Sebastian; Steih, Kristina; Urban, Karsten

doi:10.1090/mcom/3009

An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations
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by Sebastian Kestler, Kristina Steih and Karsten Urban PDF

Math. Comp. 85 (2016), 1309-1333 Request permission

Abstract:

We introduce a multitree-based adaptive wavelet Galerkin algorithm for space-time discretized linear parabolic partial differential equations, focusing on time-periodic problems. It is shown that the method converges with the best possible rate in linear complexity and can be applied for a wide range of wavelet bases. We discuss the implementational challenges arising from the Petrov-Galerkin nature of the variational formulation and present numerical results for the heat and a convection-diffusion-reaction equation.

References

Jahrul M. Alam, Nicholas K.-R. Kevlahan, and Oleg V. Vasilyev, Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations, J. Comput. Phys. 214 (2006), no. 2, 829–857. MR 2216616, DOI 10.1016/j.jcp.2005.10.009
Roman Andreev, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations, IMA J. Numer. Anal. 33 (2013), no. 1, 242–260. MR 3020957, DOI 10.1093/imanum/drs014
Hans-Joachim Bungartz and Michael Griebel, Sparse grids, Acta Numer. 13 (2004), 147–269. MR 2249147, DOI 10.1017/S0962492904000182
Albert Cohen, Wolfgang Dahmen, and Ronald DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp. 70 (2001), no. 233, 27–75. MR 1803124, DOI 10.1090/S0025-5718-00-01252-7
A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet methods. II. Beyond the elliptic case, Found. Comput. Math. 2 (2002), no. 3, 203–245. MR 1907380, DOI 10.1007/s102080010027
Albert Cohen, Wolfgang Dahmen, and Ronald Devore, Sparse evaluation of compositions of functions using multiscale expansions, SIAM J. Math. Anal. 35 (2003), no. 2, 279–303. MR 2001102, DOI 10.1137/S0036141002412070
Nabi Chegini and Rob Stevenson, Adaptive wavelet schemes for parabolic problems: sparse matrices and numerical results, SIAM J. Numer. Anal. 49 (2011), no. 1, 182–212. MR 2783222, DOI 10.1137/100800555
Nabi Chegini and Rob Stevenson, The adaptive tensor product wavelet scheme: sparse matrices and the application to singularly perturbed problems, IMA J. Numer. Anal. 32 (2012), no. 1, 75–104. MR 2875244, DOI 10.1093/imanum/drr013
Wolfgang Dahmen, Wavelet and multiscale methods for operator equations, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 55–228. MR 1489256, DOI 10.1017/S0962492900002713
Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51–150. MR 1689432, DOI 10.1017/S0962492900002816
T. Dijkema, Adaptive tensor product wavelet methods for solving PDEs, PhD thesis, Universiteit Utrecht, 2009.
Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. MR 1156075, DOI 10.1007/978-3-642-58090-1
Tammo Jan Dijkema, Christoph Schwab, and Rob Stevenson, An adaptive wavelet method for solving high-dimensional elliptic PDEs, Constr. Approx. 30 (2009), no. 3, 423–455. MR 2558688, DOI 10.1007/s00365-009-9064-0
Tsogtgerel Gantumur, Helmut Harbrecht, and Rob Stevenson, An optimal adaptive wavelet method without coarsening of the iterands, Math. Comp. 76 (2007), no. 258, 615–629. MR 2291830, DOI 10.1090/S0025-5718-06-01917-X
M. Griebel and P. Oswald, Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems, Adv. Comput. Math. 4 (1995), no. 1-2, 171–206. MR 1338900, DOI 10.1007/BF02123478
M. Griebel and D. Oeltz, A sparse grid space-time discretization scheme for parabolic problems, Computing 81 (2007), no. 1, 1–34. MR 2369419, DOI 10.1007/s00607-007-0241-3
G. Horton and S. Vandewalle, A space-time multigrid method for parabolic partial differential equations, SIAM J. Sci. Comput. 16 (1995), no. 4, 848–864. MR 1335894, DOI 10.1137/0916050
D. Jürgens, M. Palm, S. Singer, and K. Urban, Numerical optimization of the Voith-Schneider® Propeller, ZAMM 87 (2007), no. 10, 698–710.
Y. Kawajiri and L. T. Biegler, Optimization strategies for simulated moving bed and PowerFeed processes, AIChE Journal 52 (2006), no. 4, 1343–1350.
S. Kestler, On the adaptive tensor product wavelet Galerkin method with applications in finance, PhD thesis, University of Ulm, 2013.
Sebastian Kestler and Rob Stevenson, An efficient approximate residual evaluation in the adaptive tensor product wavelet method, J. Sci. Comput. 57 (2013), no. 3, 439–463. MR 3123552, DOI 10.1007/s10915-013-9712-1
Sebastian Kestler and Rob Stevenson, Fast evaluation of system matrices w.r.t. multi-tree collections of tensor product refinable basis functions, J. Comput. Appl. Math. 260 (2014), 103–116. MR 3133333, DOI 10.1016/j.cam.2013.09.015
Dominik Meidner and Boris Vexler, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control Optim. 46 (2007), no. 1, 116–142. MR 2299622, DOI 10.1137/060648994
Pál-Andrej Nitsche, Best $N$ term approximation spaces for tensor product wavelet bases, Constr. Approx. 24 (2006), no. 1, 49–70. MR 2217525, DOI 10.1007/s00365-005-0609-6
T. Raasch, Adaptive wavelet and frame schemes for elliptic and parabolic equations, PhD thesis, University Marburg, 2007.
A. Rupp, High dimensional wavelet methods for structures financial products, PhD thesis, University of Ulm, 2013.
Christoph Schwab and Rob Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78 (2009), no. 267, 1293–1318. MR 2501051, DOI 10.1090/S0025-5718-08-02205-9
Christoph Schwab and Rob Stevenson, Fast evaluation of nonlinear functionals of tensor product wavelet expansions, Numer. Math. 119 (2011), no. 4, 765–786. MR 2854127, DOI 10.1007/s00211-011-0397-9
Rob Stevenson, Adaptive wavelet methods for solving operator equations: an overview, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 543–597. MR 2648381, DOI 10.1007/978-3-642-03413-8_{1}3
Winfried Sickel and Tino Ullrich, Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross, J. Approx. Theory 161 (2009), no. 2, 748–786. MR 2563079, DOI 10.1016/j.jat.2009.01.001
Hal L. Smith and Xiao-Qiang Zhao, Dynamics of a periodically pulsed bio-reactor model, J. Differential Equations 155 (1999), no. 2, 368–404. MR 1698559, DOI 10.1006/jdeq.1998.3587
Karsten Urban and Anthony T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Math. Acad. Sci. Paris 350 (2012), no. 3-4, 203–207 (English, with English and French summaries). MR 2891112, DOI 10.1016/j.crma.2012.01.026
Karsten Urban and Anthony T. Patera, An improved error bound for reduced basis approximation of linear parabolic problems, Math. Comp. 83 (2014), no. 288, 1599–1615. MR 3194123, DOI 10.1090/S0025-5718-2013-02782-2
Karsten Urban, Wavelet methods for elliptic partial differential equations, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2009. MR 2473650
Christoph Zenger, Sparse grids, Parallel algorithms for partial differential equations (Kiel, 1990) Notes Numer. Fluid Mech., vol. 31, Friedr. Vieweg, Braunschweig, 1991, pp. 241–251. MR 1167882