# Numerical Analysis and Scientific Computing Seminar

## A Composite Spectral Method for Variable Coefficient Elliptic PDEs with its Own Fast Direct Solver

Variable coefficient elliptic PDEs arise in a variety of applications including non-destructive testing, geophysics, and designing materials. Typically finite element or spectral element methods are used to discretize the PDE resulting in a large linear system which is often solved via an iterative method (e.g. GMRES). Often the system is ill-conditioned, meaning many iterations are needed to obtain a solution. For applications with multiple right hand sides, this approach is computationally prohibitive. In this talk, we present a high-order accurate discretization technique designed for variable coefficient problems with smooth solutions. The resulting linear system is solved via a fast direct solver with $O(N)$ complexity where $N$ is the number of discretization points. Each additional solve is also $O(N)$ but with a much smaller constant. Numerical results will illustrate the performance of the method.