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

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 $quantized$ $tensor$ $train$ 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: $N = O(log^5 \epsilon^{-1})$, where $\epsilon \in (0,1)$ 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 $\epsilon$ in the energy norm with $N = O(log^K \epsilon^{-1})$ parameters, where $k<3$.