Adrianna Gillman

A Direct Solver with

A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct (as opposed to iterative) solver that has optimal

O(N)

O(N)

Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method

complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the Laplace and the Stokes equations. Numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of

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

10^{-10}

A fast direct solver for quasi-periodic scattering problems

or better for challenging problems such as highly oscillatory Helmholtz problems and convection-dominated convection-diffusion equations. In terms of speed, it is demonstrated that a problem with a nonoscillatory solution that was discretized using

A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape

10^{8}

Short note: A simplified technique for the efficient and highly accurate discretization of boundary integral equations in 2D on domains with corners

nodes can be solved in 115 minutes on a personal workstation with two quad-core 3.3 GHz CPUs. Since the solver is direct, and the “solution ...

Numerical Homogenization via Approximation of the Solution Operator

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(N2) to O(N1.5) 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., ℋ

High Resolution Inverse Scattering in Two Dimensions Using Recursive Linearization

\mathcal {H}

A fast direct solver for boundary value problems on locally perturbed geometries

-matrix arithmetic). This paper demonstrates that such 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). In this case, a modified version of the accelerated nested dissection scheme can execute any solve beyond the first in O(Nboundary) operations, where Nboundary denotes the number of points on the boundary. Typically, Nboundary ∼ N0.5. Numerical examples demonstrate the effectiveness of the procedure for a broad range of elliptic PDEs that includes both the Laplace and Helmholtz equations.