# Numerical solution of parabolic equations in high dimensions

Tobias von Petersdorff; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 38, Issue: 1, page 93-127
- ISSN: 0764-583X

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topvon Petersdorff, Tobias, and Schwab, Christoph. "Numerical solution of parabolic equations in high dimensions." ESAIM: Mathematical Modelling and Numerical Analysis 38.1 (2010): 93-127. <http://eudml.org/doc/194210>.

@article{vonPetersdorff2010,

abstract = {
We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb\{R\}^d$ and for T > 0 in
dimension dd ≥ 1. We use a wavelet based sparse grid
space discretization with mesh-width h and order pd ≥ 1, and
hp discontinuous Galerkin time-discretization of order $r =
O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many time
steps. The linear systems in each time step are solved iteratively
by $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner.
We prove that this algorithm gives an L2(Ω)-error of
O(N-p) for u(x,T) where N is the total number of operations,
provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ 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.
},

author = {von Petersdorff, Tobias, Schwab, Christoph},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Discontinuous Galerkin method; sparse grid; wavelets.; discontinuous Galerkin method; wavelets; parabolic partial differential equation; high dimension; finite elements; GMRES iterations; preconditioner; error estimates; algorithm; convergence; numerical results; heat equation},

language = {eng},

month = {3},

number = {1},

pages = {93-127},

publisher = {EDP Sciences},

title = {Numerical solution of parabolic equations in high dimensions},

url = {http://eudml.org/doc/194210},

volume = {38},

year = {2010},

}

TY - JOUR

AU - von Petersdorff, Tobias

AU - Schwab, Christoph

TI - Numerical solution of parabolic equations in high dimensions

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 38

IS - 1

SP - 93

EP - 127

AB -
We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ and for T > 0 in
dimension dd ≥ 1. We use a wavelet based sparse grid
space discretization with mesh-width h and order pd ≥ 1, and
hp discontinuous Galerkin time-discretization of order $r =
O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many time
steps. The linear systems in each time step are solved iteratively
by $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner.
We prove that this algorithm gives an L2(Ω)-error of
O(N-p) for u(x,T) where N is the total number of operations,
provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ 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.

LA - eng

KW - Discontinuous Galerkin method; sparse grid; wavelets.; discontinuous Galerkin method; wavelets; parabolic partial differential equation; high dimension; finite elements; GMRES iterations; preconditioner; error estimates; algorithm; convergence; numerical results; heat equation

UR - http://eudml.org/doc/194210

ER -

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