An Efficient Method For Solving Highly Anisotropic Elliptic Equations

TL;DR

This paper introduces a novel solution method for highly anisotropic elliptic PDEs that exploits the problem's high condition number to efficiently split and solve the equations in lower dimensions, reducing computational cost.

Contribution

The paper presents a new approach that leverages the high condition number of anisotropic elliptic operators to decompose and solve the problem more efficiently, applicable on coarse grids or with iterative methods.

Findings

Method effectively reduces the number of iterations in iterative solvers.

Applicable to a wide range of anisotropic elliptic problems with arbitrary coordinates.

Can be used standalone or with standard iterative methods.

Abstract

Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterised by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest derivative in one direction is much larger than the terms in the remaining directions, the discretized elliptic operator often has a very large condition number - taking the solution even further out of reach using traditional methods. This paper will demonstrate a solution method for such ill-behaved problems. The high condition number of the D-dimensional discretized elliptic operator will be exploited to split the problem into a series of well-behaved one and (D-1)-dimensional elliptic problems. This solution technique can be used alone on sufficiently coarse grids, or in conjunction with standard iterative methods, such as Conjugate Gradient, to substantially reduce the number of iterations needed to solve the problem to a specified accuracy. The solution is formulated analytically for a generic anisotropic problem using arbitrary coordinates, hopefully bringing this method into the scope of a wide variety of applications.