Schwab, Christoph and Suli, Endre and Todor, RaduAlexandru (2007) Sparse finite element approximation of highdimensional transportdominated diffusion problems. Technical Report. Unspecified. (Submitted)

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Abstract
Partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the exponential growth of computational complexity as a function of the dimension d of the problem domain, the socalled ``curse of dimension'', is exacerbated by the fact that the problem may be transportdominated. We develop the numerical analysis of stabilized sparse tensorproduct finite element methods for such highdimensional, nonselfadjoint and possibly degenerate secondorder partial differential equations, using piecewise polynomials of degree p > 0. Our convergence analysis is based on new highdimensional approximation results in sparse tensorproduct spaces. By tracking the dependence of the various constants on the dimension and the polynomial degree p, we show in the case of elliptic transportdominated diffusion problems that for p > 0 the error constant exhibits exponential decay as d tends to infinity. In the general case when the characteristic form of the partial differential equation is nonnegative, under a mild condition relating p to d, the error constant is shown to grow no faster than quadratically in d. In any case, the sparse stabilized finite element method exhibits an optimal rate of convergence with respect to the meshsize, up to a factor that is polylogarithmic in the meshsize.
Dedicated to Henryk Wozniakowski, on the occasion of his 60th birthday.
Item Type:  Technical Report (Technical Report) 

Subjects:  H  N > Numerical analysis 
Research Groups:  Numerical Analysis Group 
ID Code:  1098 
Deposited By:  Lotti Ekert 
Deposited On:  07 May 2011 08:02 
Last Modified:  29 May 2015 18:47 
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