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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: $$N={\mathcal {O}} ( \log ^{5} \varepsilon ^{-1} ) $$N=O(log5ε-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 $$N={\mathcal {O}} ( \log ^{\kappa } \varepsilon ^{-1} ) $$N=O(logκε-1) parameters, where $$\kappa <3$$κ<3.

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