A hybrid interpolation ACA accelerated method for parabolic boundary integral operators

TL;DR

This paper introduces a hybrid interpolation ACA accelerated method for efficiently solving dense linear systems arising from boundary integral reformulations of the heat equation, combining Chebyshev interpolation and adaptive cross approximation.

Contribution

It develops an ACA framework tailored for parabolic boundary integral operators, analyzing the ACA tolerance adjustment for temporal separation and demonstrating its effectiveness.

Findings

The method achieves data-sparse matrix approximations with controlled accuracy.

Numerical results confirm the theoretical error estimates.

The approach accelerates the solution of boundary integral equations for heat problems.

Abstract

We consider piecewise polynomial discontinuous Galerkin discretizations of boundary integral reformulations of the heat equation. The resulting linear systems are dense and block-lower triangular and hence can be solved by block forward elimination. For the fast evaluation of the history part, the matrix is subdivided into a family of sub-matrices according to the temporal separation. Separated blocks are approximated by Chebyshev interpolation of the heat kernel in time. For the spatial variable, we propose an adaptive cross approximation (ACA) framework to obtain a data-sparse approximation of the entire matrix. We analyse how the ACA tolerance must be adjusted to the temporal separation and present numerical results for a benchmark problem to confirm the theoretical estimates.