Numerical solution of parabolic equations in high dimensions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, p. 93-127

We consider the numerical solution of diffusion problems in $\left(0,T\right)×\Omega$ for $\Omega \subset {ℝ}^{d}$ and for $T>0$ in dimension $d\ge 1$. We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p\ge 1$, and $hp$ discontinuous Galerkin time-discretization of order $r=O\left(\left|logh\right|\right)$ on a geometric sequence of $O\left(\left|logh\right|\right)$ many time steps. The linear systems in each time step are solved iteratively by $O\left(\left|logh\right|\right)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an ${L}^{2}\left(\Omega \right)$-error of $O\left({N}^{-p}\right)$ for $u\left(x,T\right)$ where $N$ is the total number of operations, provided that the initial data satisfies ${u}_{0}\in {H}^{ϵ}\left(\Omega \right)$ with $ϵ>0$ and that $u\left(x,t\right)$ is smooth in $x$ for $t>0$. Numerical experiments in dimension $d$ up to $25$ confirm the theory.

DOI : https://doi.org/10.1051/m2an:2004005
Classification:  65N30
Keywords: discontinuous Galerkin method, sparse grid, wavelets
@article{M2AN_2004__38_1_93_0,
author = {Petersdorff, Tobias Von and Schwab, Christoph},
title = {Numerical solution of parabolic equations in high dimensions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
pages = {93-127},
doi = {10.1051/m2an:2004005},
zbl = {1083.65095},
mrnumber = {2073932},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_1_93_0}
}

Petersdorff, Tobias Von; Schwab, Christoph. Numerical solution of parabolic equations in high dimensions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, pp. 93-127. doi : 10.1051/m2an:2004005. http://www.numdam.org/item/M2AN_2004__38_1_93_0/

[1] H. Amann, Linear and Quasilinear Parabolic Problems 1: Abstract Linear Theory. Birkhäuser, Basel (1995). | MR 1345385 | Zbl 0819.35001

[2] H.-J. Bungartz and M. Griebel, A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives. J. Complexity 15 (1999) 167-199. | Zbl 0954.65078

[3] S.C. Eisenstat, H.C. Elman and M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20 (1983) 345-357. | Zbl 0524.65019

[4] M. Griebel and S. Knapek, Optimized tensor product approximation spaces. Constr. Approx. 16 (2000) 525-540. | Zbl 0969.65107

[5] M. Griebel, P. Oswald and T. Schiekofer, Sparse grids for boundary integral equations. Numer. Math. 83 (1999) 279-312. | Zbl 0935.65131

[6] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I. Springer-Verlag (1972). | Zbl 0223.35039

[7] P. Oswald, On best N-term approximation by Haar functions in ${H}^{s}$-norms, in Metric Function Theory and Related Topics in Analysis. S.M. Nikolskij, B.S. Kashin, A.D. Izaak Eds., AFC, Moscow (1999) 137-163 (in Russian).

[8] H.C. Öttinger, Stochastic Processes in polymeric fluids. Springer-Verlag (1998). | MR 1383323 | Zbl 0995.60098

[9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci., Springer-Verlag, New York 44 (1983). | MR 710486 | Zbl 0516.47023

[10] G. Schmidlin, C. Lage and C. Schwab, Rapid solution of first kind boundary integral equations in ${ℝ}^{3}$. Eng. Anal. Bound. Elem. 27 (2003) 469-490. | Zbl 1038.65129

[11] D. Schötzau, hp-DGFEM for Parabolic Evolution Problems. Dissertation ETH Zurich (1999).

[12] D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Analysis 38 (2000) 837-875. | Zbl 0978.65091

[13] D. Schötzau and C. Schwab, $hp$-Discontinuous Galerkin time-stepping for parabolic problems. C.R. Acad. Sci. Paris 333 (2001) 1121-1126. | Zbl 0993.65108

[14] C. Schwab, $p$ and $hp$ Finite Element Methods. Oxford University Press (1998). | MR 1695813 | Zbl 0910.73003

[15] C. Schwab and R.A. Todor, Sparse finite elements for stochastic elliptic problems-higher order moments (in press in Computing 2003), http://www.math.ethz.ch/research/groups/sam/reports/2003 | MR 2009650 | Zbl 1044.65006

[16] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag (1997). | MR 1479170 | Zbl 0884.65097

[17] T. Von Petersdorff and C. Schwab, Wavelet-discretizations of parabolic integro-differential equations. SIAM J. Numer. Anal. 41 (2003) 159-180. | Zbl 1050.65134

[18] T. Werder, D. Schötzau, K. Gerdes and C. Schwab, $hp$-Discontinuous Galerkin time-stepping for parabolic problems. Comput. Methods Appl. Mech. Eng. 190 (2001) 6685-6708. | Zbl 0992.65103