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

摘要

The large sparse linear systems arising from the finite element or finitedifference discretization of elliptic PDEs can be solved directly via, e.g.,nested dissection or multifrontal methods. Such techniques reorder the nodes inthe 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 hasrecently 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 thatsuch \textit{accelerated} nested dissection techniques become particularlyeffective for boundary value problems without body loads when the solution issought for several different sets of boundary data, and the solution isrequired only near the boundary (as happens, e.g., in the computationalmodeling of scattering problems, or in engineering design of linearly elasticsolids.