Numerical solution of parabolic equations in high dimensions

Petersdorff, Tobias Von ; Schwab, Christoph

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 38 (2004) no. 1, pp. 93-127

Résumé

We consider the numerical solution of diffusion problems in $(0,T)\times \Omega$ 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 $u(x,T)$ 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 $u(x,t)$ is smooth in $x$ for $t>0$. Numerical experiments in dimension $d$ up to $25$ confirm the theory.

Détail

DOI : 10.1051/m2an:2004005

Classification : 65N30
Keywords: discontinuous Galerkin method, sparse grid, wavelets