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Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions

by V. Kazeev and Ch. Schwab

(Report number 2015-24)

Abstract
We analyze the approximation of the solutions of second-order elliptic problems, which have point singularities but belong to a countably normed space of analytic functions, with a first-order, $h$-version finite element (FE) method based on uniform tensor-product meshes. The FE solutions are well known to converge with algebraic rate at most $1/2$ in terms of the number of degrees of freedom, and even slower in the presence of singularities. We analyze the compression of the FE coefficient vectors represented in the so-called \emph{quantized tensor train} format. We prove, in a reference square, that the corresponding FE approximations converge exponentially in terms of the effective number $N$ of degrees of freedom involved in the representation: $N=\mathcal{O}(\log^{5} \varepsilon^{-1})$, where $\varepsilon\in (0,1)$ is the accuracy measured in the energy norm. Numerically we show for solutions from the same class that the entire process of solving the tensor-structured Galerkin first-order FE discretization can achieve accuracy $\varepsilon$ in the energy norm with $N=\mathcal{O}(\log^{\kappa} \varepsilon^{-1})$ parameters, where $\kappa<3$.

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

@Techreport{KS15_614,
  author = {V. Kazeev and Ch. Schwab},
  title = {Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-24.pdf },
  year = {2015}
}

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