Numerical solution of parabolic equations in high dimensions

Petersdorff, Tobias Von; Schwab, Christoph

doi:10.1051/m2an:2004005

Petersdorff, Tobias Von ; Schwab, Christoph

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 93-127.

Résumé

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

MR Zbl | 6 citations dans Numdam

DOI : 10.1051/m2an:2004005

Classification : 65N30
Mots-clés : discontinuous Galerkin method, sparse grid, wavelets

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     author = {Petersdorff, Tobias Von and Schwab, Christoph},
     title = {Numerical solution of parabolic equations in high dimensions},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {93--127},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {1},
     year = {2004},
     doi = {10.1051/m2an:2004005},
     mrnumber = {2073932},
     zbl = {1083.65095},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2004005/}
}

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Petersdorff, Tobias Von; Schwab, Christoph. Numerical solution of parabolic equations in high dimensions. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 93-127. doi : 10.1051/m2an:2004005. http://www.numdam.org/articles/10.1051/m2an:2004005/

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