Sparse finite element approximation of high-dimensional transport-dominated diffusion problems

Christoph Schwab; Endre Süli; Radu Alexandru Todor

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 5, page 777-819
  • ISSN: 0764-583X

Abstract

top
We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form - a : ∇ ∇ u + b · ∇ u + c u = f ( x ) , x ∈ Ω = ( 0 , 1 ) d ⊂ ℝ d , where a ∈ ℝ d × d is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm | | | · | | | SD , provided u belongs to the space ℋ k + 1 ( Ω ) of functions with square-integrable mixed (k+1)st derivatives: | | | u - u h | | | SD ≤ C p , t d 2 max { ( 2 - p ) + , κ 0 d - 1 , κ 1 d } ( | a | h L t + | b | 1 2 h L t + 1 2 + c 1 2 h L t + 1 ) | u | ℋ t + 1 ( Ω ) , where κ i = κ i ( p , t , L ) , i=0,1, and 1 ≤ t ≤ min ( k , p ) . We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems κ 0 , κ 1 ∈ ( 0 , 1 ) , and hence for p ≥ 1 the 'error constant' C p , t d 2 max { ( 2 - p ) + , κ 0 d - 1 , κ 1 d } exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than 𝒪 ( d 2 ) . In any case, in the absence of assumptions that relate L, p and d, the error | | | u - u h | | | SD is still bounded by κ * d - 1 | log 2 h L | d - 1 𝒪 ( | a | h L t + | b | 1 2 h L t + 1 2 + c 1 2 h L t + 1 ) , where κ * ∈ ( 0 , 1 ) for all L, p, d ≥ 2.

How to cite

top

Schwab, Christoph, Süli, Endre, and Todor, Radu Alexandru. "Sparse finite element approximation of high-dimensional transport-dominated diffusion problems." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 777-819. <http://eudml.org/doc/250411>.

@article{Schwab2008,
abstract = { We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$, $x \in \Omega = (0,1)^d \subset \mathbb\{R\}^d$, where $a \in \mathbb\{R\}^\{d\times d\}$ is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm $|||\cdot|||_\{\rm SD\}$, provided u belongs to the space $\mathcal\{H\}^\{k+1\}(\Omega)$ of functions with square-integrable mixed (k+1)st derivatives: \[ |||u-u\_h|||\_\{\rm SD\}\leq C\_\{p,t\} d^2 \max\\{(2-p)\_+,\kappa\_0^\{d-1\},\kappa\_1^d\\} (|\sqrt\{a\}| h\_L^t + |b|^\{\frac\{1\}\{2\}\} h\_L^\{t+\frac\{1\}\{2\}\} + c^\{\frac\{1\}\{2\}\} h\_L^\{t+1\} \!)|u|\_\{\mathcal\{H\}^\{t+1\}(\Omega)\}, \qquad \qquad \qquad \] where $\kappa_i=\kappa_i(p,t,L)$, i=0,1, and $1 \leq t \leq \min(k,p)$. We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems $\kappa_0, \kappa_1 \in (0,1)$, and hence for p ≥ 1 the 'error constant' $C_\{p,t\} d^2 \max\\{(2-p)_+,\kappa_0^\{d-1\},\kappa_1^d\\}$ exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than $\mathcal\{O\}(d^2)$. In any case, in the absence of assumptions that relate L, p and d, the error $|||u - u_h|||_\{\rm SD\}$ is still bounded by $\kappa_\ast^\{d-1\} |\log_2 h_L|^\{d-1\}\mathcal\{O\}(|\sqrt\{a\}| h_L^t + |b|^\{\frac\{1\}\{2\}\} h_L^\{t+\frac\{1\}\{2\}\} + c^\{\frac\{1\}\{2\}\} h_L^\{t+1\})$, where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2. },
author = {Schwab, Christoph, Süli, Endre, Todor, Radu Alexandru},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {High-dimensional Fokker-Planck equations; partial differential equations with nonnegative characteristic form; sparse finite element method.; high-dimensional Fokker-Planck equations; non-negative characteristic form; sparse finite element method; error bounds; convergence; sparse tensor-product spaces},
language = {eng},
month = {7},
number = {5},
pages = {777-819},
publisher = {EDP Sciences},
title = {Sparse finite element approximation of high-dimensional transport-dominated diffusion problems},
url = {http://eudml.org/doc/250411},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Schwab, Christoph
AU - Süli, Endre
AU - Todor, Radu Alexandru
TI - Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 5
SP - 777
EP - 819
AB - We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$, $x \in \Omega = (0,1)^d \subset \mathbb{R}^d$, where $a \in \mathbb{R}^{d\times d}$ is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm $|||\cdot|||_{\rm SD}$, provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functions with square-integrable mixed (k+1)st derivatives: \[ |||u-u_h|||_{\rm SD}\leq C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\} (|\sqrt{a}| h_L^t + |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1} \!)|u|_{\mathcal{H}^{t+1}(\Omega)}, \qquad \qquad \qquad \] where $\kappa_i=\kappa_i(p,t,L)$, i=0,1, and $1 \leq t \leq \min(k,p)$. We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems $\kappa_0, \kappa_1 \in (0,1)$, and hence for p ≥ 1 the 'error constant' $C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\}$ exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than $\mathcal{O}(d^2)$. In any case, in the absence of assumptions that relate L, p and d, the error $|||u - u_h|||_{\rm SD}$ is still bounded by $\kappa_\ast^{d-1} |\log_2 h_L|^{d-1}\mathcal{O}(|\sqrt{a}| h_L^t + |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1})$, where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2.
LA - eng
KW - High-dimensional Fokker-Planck equations; partial differential equations with nonnegative characteristic form; sparse finite element method.; high-dimensional Fokker-Planck equations; non-negative characteristic form; sparse finite element method; error bounds; convergence; sparse tensor-product spaces
UR - http://eudml.org/doc/250411
ER -

References

top
  1. K. Babenko, Approximation by trigonometric polynomials is a certain class of periodic functions of several variables. Soviet Math. Dokl.1 (1960) 672–675. Russian original in Dokl. Akad. Nauk SSSR132 (1960) 982–985.  
  2. J.W. Barrett and E. Süli, Existence of global weak solutions to kinetic models of dilute polymers. Multiscale Model. Simul.6 (2007) 506–546.  
  3. J.W. Barrett, C. Schwab and E. Süli, Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci.15 (2005) 939–983.  
  4. R.F. Bass, Diffusion and Elliptic Operators. Springer-Verlag, New York (1997).  
  5. T.S. Blyth and E.F. Robertson, Further Linear Algebra. Springer-Verlag, London (2002).  
  6. H.-J. Bungartz, Finite elements of higher order on sparse grids. Habilitation thesis, Informatik, TU München, Aachen: Shaker Verlag (1998).  
  7. H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numer.13 (2004) 1–123.  
  8. R. DeVore, S. Konyagin and V. Temlyakov, Hyperbolic wavelet approximation. Constr. Approx.14 (1998) 1–26.  
  9. J. Dick, I.H. Sloan, X. Wang and H. Woźniakowski, Good lattice rules in weighted Korobov spaces with general weights. Numer. Math.103 (2006) 63–97.  
  10. J. Elf, P. Lötstedt and P. Sjöberg, Problems of high dimension in molecular biology, in Proceedings of the 19th GAMM-Seminar Leipzig, W. Hackbusch Ed. (2003) 21–30.  
  11. M. Griebel, Sparse grids and related approximation schemes for higher dimensional problems, in Foundations of Computational Mathematics 2005, L.-M. Pardo, A. Pinkus, E. Süli, M. Todd Eds., Cambridge University Press (2006) 106–161.  
  12. V.H. Hoang and C. Schwab, High dimensional finite elements for elliptic problems with multiple scales. Multiscale Model. Simul.3 (2005) 168–194.  
  13. L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients. Springer-Verlag, Berlin, Reprint of the 1983 edition (2005).  
  14. P. Houston and E. Süli, Stabilized hp-finite element approximation of partial differential equations with non-negative characteristic form. Computing66 (2001) 99–119.  
  15. P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal.39 (2002) 2133–2163.  
  16. B. Lapeyre, É. Pardoux and R. Sentis, Introduction to Monte-Carlo Methods for Transport and Diffusion Equations, Oxford Texts in Applied and Engineering Mathematics. Oxford University Press, Oxford (2003).  
  17. P. Laurençot and S. Mischler, The continuous coagulation fragmentation equations with diffusion. Arch. Rational Mech. Anal.162 (2002) 45–99.  
  18. C. Le Bris and P.-L. Lions, Renormalized solutions of some transport equations with W1,1 velocities and applications. Annali di Matematica183 (2004) 97–130.  
  19. E. Novak and K. Ritter, The curse of dimension and a universal method for numerical integration, in Multivariate Approximation and Splines, G. Nürnberger, J. Schmidt and G. Walz Eds., International Series in Numerical Mathematics, Birkhäuser, Basel (1998) 177–188.  
  20. O.A. Oleĭnik and E.V. Radkevič, Second Order Equations with Nonnegative Characteristic Form. American Mathematical Society, Providence, RI (1973).  
  21. H.-C. Öttinger, Stochastic Processes in Polymeric Fluids. Springer-Verlag, New York (1996).  
  22. H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics24. Springer-Verlag, New York (1996).  
  23. C. Schwab, p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Numerical Methods and Scientific Computation. Clarendon Press, Oxford (1998).  
  24. S. Smolyak, Quadrature and interpolation formulas for products of certain classes of functions. Soviet Math. Dokl.4 (1963) 240–243. Russian original in Dokl. Akad. Nauk SSSR148 (1963) 1042–1045.  
  25. E. Süli, Finite element approximation of high-dimensional transport-dominated diffusion problems, in Foundations of Computational Mathematics 2005, L.-M. Pardo, A. Pinkus, E. Süli, M. Todd Eds., Cambridge University Press (2006) 343–370. Available at: http://web.comlab.ox.ac.uk/oucl/publications/natr/index.html  
  26. E. Süli, Finite element algorithms for transport-diffusion problems: stability, adaptivity, tractability, in Invited Lecture at the International Congress of Mathematicians, Madrid, 22–30 August 2006. Available at: http://web.comlab.ox.ac.uk/work/endre.suli/Suli-ICM2006.pdf  
  27. V. Temlyakov, Approximation of functions with bounded mixed derivative, in Proc. Steklov Inst. of Math.178, American Mathematical Society, Providence, RI (1989).  
  28. N.G. van Kampen, Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (1992).  
  29. T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high dimensions. ESAIM: M2AN38 (2004) 93–128.  
  30. G. Wasilkowski and H. Woźniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complexity11 (1995) 1–56.  
  31. C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations, W. Hackbusch Ed., Notes on Numerical Fluid Mechanics31, Vieweg, Braunschweig/Wiesbaden (1991).  
  32. G.W. Zumbusch, A sparse grid PDE solver, in Advances in Software Tools for Scientific Computing, H.P. Langtangen, A.M. Bruaset and E. Quak Eds., Lecture Notes in Computational Science and Engineering10, Springer, Berlin (Proceedings SciTools '98) (2000) 133–177.  

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.