An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads

TL;DR

This paper presents an O(N) algorithm for efficiently constructing the solution operator for 2D elliptic boundary value problems without body loads, especially effective when multiple boundary conditions are involved.

Contribution

It introduces an accelerated nested dissection method leveraging matrix structure to achieve linear complexity for specific boundary value problems.

Findings

Achieves linear O(N) complexity for solution operator construction.

Effective for multiple boundary conditions and boundary-near solutions.

Reduces computational cost in scattering and elastic solid modeling.

Abstract

The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid to reduce the asymptotic complexity of Gaussian elimination from $O(N^{2})$ to $O(N^{1.5})$ for typical problems in two dimensions. It has recently been demonstrated that the complexity can be further reduced to O(N) by exploiting structure in the dense matrices that arise in such computations (using, e.g., $\mathcal{H}$-matrix arithmetic). This paper demonstrates that such \textit{accelerated} nested dissection techniques become particularly effective for boundary value problems without body loads when the solution is sought for several different sets of boundary data, and the solution is required only near the boundary (as happens, e.g., in the computational modeling of scattering problems, or in engineering design of linearly elastic solids.