Superconvergence of immersed finite volume methods for one-dimensional interface problems

TL;DR

This paper introduces high-order immersed finite volume methods for 1D interface problems, demonstrating optimal convergence and superconvergence properties, with numerical results confirming theoretical predictions.

Contribution

The paper develops and analyzes high-order immersed finite volume methods with proven superconvergence for 1D interface problems, extending finite volume theory.

Findings

Optimal convergence in H1 and L2 norms.

Superconvergence of order p+2 at Lobatto points.

Flux superconvergence of order p+1 at Gauss points.

Abstract

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in H1 and L2 norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order p+2 at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order p+1 at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gaussian points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.