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

Kazeev, Vladimir ; Schwab, Christoph

Numerische Mathematik, 2018, Vol.138(1), pp.133-190 [Peer Reviewed Journal]

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  • Title:
    Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions
  • Author/Creator: Kazeev, Vladimir ; Schwab, Christoph
  • Language: English
  • Subjects: 15A69 ; 35C99 ; 35J25 ; 65N12 ; 65N30 ; 65N35
  • Is Part Of: Numerische Mathematik, 2018, Vol.138(1), pp.133-190
  • Description: We analyze the approximation of the solutions of second-order elliptic problems, which have point singularities and 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 quantized-tensor-train format. We prove, in a reference square, that the resulting FE approximations converge exponentially in terms of the effective number N of degrees of freedom involved in the representation: ={\mathcal {O}} ( \log ^{5} \varepsilon ^{-1} ) N = O ( log 5 ε - 1 ) , where varepsilon \in (0,1) ε ∈ ( 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 ε in the energy norm with ={\mathcal {O}} ( \log ^{\kappa } \varepsilon ^{-1} ) N = O ( log κ ε - 1 ) parameters, where kappa <3 κ < 3 .
  • Identifier: ISSN: 0029-599X ; E-ISSN: 0945-3245 ; DOI: 10.1007/s00211-017-0899-1