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Approximation of Singularities by Quantized-Tensor FEM

by V. Kazeev and Ch. Schwab

(Report number 2015-16)

Abstract
In $d$ dimensions, first-order tensor-product finite-element (FE) approximations of the solutions of second-order elliptic problems are well known to converge algebraically, with rate at most $1/d$ in the energy norm and with respect to the number of degrees of freedom. On the other hand, FE methods of higher regularity may achieve exponential convergence, e.g. global spectral methods for analytic solutions and $hp$ methods for solutions from certain countably normed spaces, which may exhibit singularities. In this note, we revisit, in one dimension, the tensor-structured approach to the $h$-FE approximation of singular functions. We outline a proof of the exponential convergence of such approximations represented in the quantized-tensor-train (QTT) format. Compared to special approximation techniques, such as $hp$, that approach is fully adaptive in the sense that it finds suitable approximation spaces algorithmically. The convergence is measured with respect to the number of parameters used to represent the solution, which is not the dimension of the first-order FE space, but depends only polylogarithmically on that. We demonstrate the convergence numerically for a simple model problem and find the rate to be approximately the same as for $hp$ approximations.

Keywords: Numerical analysis, singular solution, analytic regularity, finite-element method, tensor decomposition, low rank, tensor rank, multilinear algebra, tensor train

@Techreport{KS15_606,
  author = {V. Kazeev and Ch. Schwab},
  title = {Approximation of Singularities by Quantized-Tensor FEM},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-16.pdf },
  year = {2015}
}

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