Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems

TL;DR

This paper introduces a second order accurate multigrid method for solving one-dimensional elliptic equations with discontinuous coefficients at interfaces, utilizing ghost points and iterative interface condition relaxation.

Contribution

It presents a novel second order multigrid approach for 1D elliptic problems with discontinuous coefficients, based on the Ghost Fluid Method and multi-domain formulation.

Findings

Achieves second order accuracy for the solution and its first derivative.

Converges efficiently with a rate close to smooth coefficient cases.

Independent of the magnitude of coefficient jumps.

Abstract

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations, leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway.