# Numerical solution of parabolic equations in high dimensions

Tobias Von Petersdorff; Christoph Schwab

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

## Access Full Article

top## Abstract

top## How to cite

topPetersdorff, Tobias Von, and Schwab, Christoph. "Numerical solution of parabolic equations in high dimensions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.1 (2004): 93-127. <http://eudml.org/doc/246004>.

@article{Petersdorff2004,

abstract = {We consider the numerical solution of diffusion problems in $(0,T) \times \Omega $ for $\Omega \subset \mathbb \{R\}^d$ 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 $r = O(\left|\log h\right|)$ 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 $L^2(\Omega )$-error of $O(N^\{-p\})$ for $u(x,T)$ where $N$ is the total number of operations, provided that the initial data satisfies $u_0 \in H^\epsilon (\Omega )$ with $\epsilon >0$ and that $u(x,t)$ is smooth in $x$ for $t>0$. Numerical experiments in dimension $d$ up to $25$ confirm the theory.},

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

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

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

language = {eng},

number = {1},

pages = {93-127},

publisher = {EDP-Sciences},

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

url = {http://eudml.org/doc/246004},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Petersdorff, Tobias Von

AU - Schwab, Christoph

TI - Numerical solution of parabolic equations in high dimensions

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 1

SP - 93

EP - 127

AB - We consider the numerical solution of diffusion problems in $(0,T) \times \Omega $ for $\Omega \subset \mathbb {R}^d$ 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 $r = O(\left|\log h\right|)$ 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 $L^2(\Omega )$-error of $O(N^{-p})$ for $u(x,T)$ where $N$ is the total number of operations, provided that the initial data satisfies $u_0 \in H^\epsilon (\Omega )$ with $\epsilon >0$ 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; parabolic partial differential equation; high dimension; finite elements; GMRES iterations; preconditioner; error estimates; algorithm; convergence; numerical results; heat equation

UR - http://eudml.org/doc/246004

ER -

## References

top- [1] H. Amann, Linear and Quasilinear Parabolic Problems 1: Abstract Linear Theory. Birkhäuser, Basel (1995). Zbl0819.35001MR1345385
- [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. Zbl0954.65078
- [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. Zbl0524.65019
- [4] M. Griebel and S. Knapek, Optimized tensor product approximation spaces. Constr. Approx. 16 (2000) 525–540. Zbl0969.65107
- [5] M. Griebel, P. Oswald and T. Schiekofer, Sparse grids for boundary integral equations. Numer. Math. 83 (1999) 279–312. Zbl0935.65131
- [6] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I. Springer-Verlag (1972). Zbl0223.35039
- [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). Zbl0995.60098MR1383323
- [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci., Springer-Verlag, New York 44 (1983). Zbl0516.47023MR710486
- [10] G. Schmidlin, C. Lage and C. Schwab, Rapid solution of first kind boundary integral equations in ${\mathbb{R}}^{3}$. Eng. Anal. Bound. Elem. 27 (2003) 469–490. Zbl1038.65129
- [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. Zbl0978.65091
- [13] D. Schötzau and C. Schwab, $hp$-Discontinuous Galerkin time-stepping for parabolic problems. C.R. Acad. Sci. Paris 333 (2001) 1121–1126. Zbl0993.65108
- [14] C. Schwab, $p$ and $hp$ Finite Element Methods. Oxford University Press (1998). Zbl0910.73003MR1695813
- [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 Zbl1044.65006MR2009650
- [16] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag (1997). Zbl0884.65097MR1479170
- [17] T. von Petersdorff and C. Schwab, Wavelet-discretizations of parabolic integro-differential equations. SIAM J. Numer. Anal. 41 (2003) 159–180. Zbl1050.65134
- [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. Zbl0992.65103

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.