A multidomain spectral method for solving elliptic equations

TL;DR

This paper introduces a flexible multidomain spectral solver for nonlinear elliptic PDEs, capable of handling complex geometries and providing superior accuracy and efficiency compared to finite difference methods.

Contribution

The paper presents a novel spectral method that unifies PDE solving, boundary conditions, and domain matching into a single system, supporting various domain types and adaptable to different PDEs.

Findings

The spectral solver outperforms finite difference codes in runtime.

The method achieves higher accuracy for smooth problems.

It successfully solves equations from general relativity initial value problems.

Abstract

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.