# 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 -

## References

top- H. Amann, Linear and Quasilinear Parabolic Problems 1: Abstract Linear Theory. Birkhäuser, Basel (1995).
- H.-J. Bungartz and M. Griebel, A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives. J. Complexity15 (1999) 167–199.
- 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.
- M. Griebel and S. Knapek, Optimized tensor product approximation spaces. Constr. Approx.16 (2000) 525–540.
- M. Griebel, P. Oswald and T. Schiekofer, Sparse grids for boundary integral equations. Numer. Math.83 (1999) 279–312.
- J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I. Springer-Verlag (1972).
- P. Oswald, On best N-term approximation by Haar functions in Hs-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).
- H.C. Öttinger, Stochastic Processes in polymeric fluids. Springer-Verlag (1998).
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci., Springer-Verlag, New York 44 (1983).
- 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.
- D. Schötzau, hp-DGFEM for Parabolic Evolution Problems. Dissertation ETH Zurich (1999).
- D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Analysis38 (2000) 837–875.
- D. Schötzau and C. Schwab, hp-Discontinuous Galerkin time-stepping for parabolic problems. C.R. Acad. Sci. Paris333 (2001) 1121–1126.
- C. Schwab, p and hp Finite Element Methods. Oxford University Press (1998).
- C. Schwab and R.A. Todor, Sparse finite elements for stochastic elliptic problems-higher order moments (in press in Computing 2003), URIhttp://www.math.ethz.ch/research/groups/sam/reports/2003
- V. Thomee, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag (1997).
- T. von Petersdorff and C. Schwab, Wavelet-discretizations of parabolic integro-differential equations. SIAM J. Numer. Anal.41 (2003) 159–180.
- 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.

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