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

by Ch. Schwab and E. Süli and R. A. Todor

(Report number 2007-04)

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 so-called "curse of dimension", is exacerbated by the fact that the problem may be transport-dominated. We develop the numerical analysis of stabilized sparse tensor product finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations, using piecewise polynomials of degree p>1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. By tracking the dependence of the various constants on the dimension d and the polynomial degree p, we show in the case of elliptic transport-dominated diffusion problems that for p>1 the error-constant exhibits exponential decay as $d \rightarrow \infty$. In the general case when the characteristic form of the partial differential equation is non-negative, under a mild condition relating $p$ to d, the error constant is shown to grow no faster than $\mathcal {O}(d^2)$. In any case, the sparse stabilized finite element method exhibits an optimal rate of convergence with respect to the mesh size $h_L$, up to a factor that is polylogarithmic in $h_L$.

Keywords: high-dimensional Fokker-Planck equations, partial differential equations with nonnegative characteristic form, sparse finite element method

@Techreport{SST07_365,
  author = {Ch. Schwab and E. S\"uli and R. A. Todor},
  title = {Sparse finite element approximation of high-dimensional transport-dominated diffusion problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2007-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2007/2007-04.pdf },
  year = {2007}
}

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