Numerical Methods for Linear Diffusion Equations in the Presence of an Interface

TL;DR

This paper develops and analyzes numerical methods, including immersed interface finite element and Euler-Maruyama schemes, for solving linear diffusion equations with discontinuous coefficients and interface conditions, applicable to both deterministic and stochastic models.

Contribution

It introduces a unified approach to handle various interface conditions in diffusion equations using immersed interface methods and stochastic schemes, with proven convergence.

Findings

Constructed immersed interface finite element methods for the deterministic problem.

Developed Euler-Maruyama scheme for the stochastic differential equation.

Proved convergence estimates for the stochastic Euler scheme.

Abstract

We consider numerical methods for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients defined by a one parameter family of interface conditions at the discontinuity. We construct immersed interface finite element methods for an alternative formulation of the original deterministic diffusion problem in which the interface condition is recast as a natural condition on the interfacial flux for which the given operator is self adjoint. An Euler-Maruyama method is developed for the stochastic differential equation corresponding to the alternative divergence formulation of the equation having a discontinuous coefficient and a one-parameter family of interface conditions. We then prove convergence estimates for the Euler scheme. The main goal is to develop numerical schemes that can accommodate specification of any one of the possible interface conditions, and to illustrate the implementation for each of the deterministic and stochastic formulations, respectively. The issues pertaining to speed-ups of the numerical schemes are left to future work.