Superconvergence of immersed finite volume methods for one-dimensional interface problems
For researchers solving interface problems with finite volume methods, this work extends superconvergence theory to immersed methods, matching standard finite volume results.
This paper develops high-order immersed finite volume methods for 1D interface problems, proving optimal convergence and superconvergence: solution accuracy of order p+2 at generalized Lobatto points and flux accuracy of order p+1 at generalized Gauss points, with rates up to 2p for diffusion problems.
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.