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Superconvergence of immersed finite element methods for interface problems Waixiang Cao1 · Xu Zhang2

· Zhimin Zhang1,3

Received: 14 November 2015 / Accepted: 30 November 2016 © Springer Science+Business Media New York 2017

Abstract In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.

This work is supported in part by the China Postdoctoral Science Foundation 2015M570026, and the National Natural Science Foundation of China (NSFC) under grants No. 91430216, 11471031, 11501026, and the US National Science Foundation (NSF) through grant DMS-1419040. Communicated by: Jan Hesthaven  Xu Zhang

[email protected] Waixiang Cao [email protected] Zhimin Zhang [email protected] 1

Beijing Computational Science Research Center, Beijing, 100193, China

2

Department of Mathematics and Statistics, Mississippi State University, Mississippi State MS 39762, USA

3

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

W. Cao et al.

Keywords Superconvergence · Immersed finite element method · Interface problems · Generalized orthogonal polynomials Mathematics Subject Classification (2010) 65N30 · 65N15 · 35R05

1 Introduction Immersed finite element (IFE) methods are a class of finite element methods (FEM) for solving differential equations with discontinuous coefficients, often known as interface problems. Unlike the classical FEM whose mesh is required to be aligned with the interface, IFE methods do not have such restriction. Consequently, IFE methods can use more structured, or even uniform meshes to solve interface problem regardless of interface location. This flexibility is advantageous for problems with complicated interfacial geometry [37] or for dynamic simulation involving a moving interface [22, 28, 29]. The main idea of IFE methods is to adapt approximating functions instead of meshes to fit the interface. On elements containing (part of) the interface, which we call interface elements, universal polynomials cannot approximate the solution accurately because of the low regularity of solution at the interface. A simple remedy is to construct piecewise polynomials as basis functions on interface elements in order to mimic the exact solution. The first IFE method was developed by Li [25] for solving the one-dimensional two-point boundary value problem. Piecewise linear shape functions were constructed on interface elements to incorporate the interface jump