title: MIB Galerkin method for elliptic interface problems.
abstract: Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the designed second order convergence of the MIB Galerkin method in theL∞andL2errors. Some of the best results are obtained in the present work when the interface isC1or Lipschitz continuous and the solution isC2continuous.
authors.affiliation: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA ; Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China.
authors.forename: Kelin
authors.initials: K
authors.lastname: Xia
authors.affiliation: Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China.
authors.forename: Meng
authors.initials: M
authors.lastname: Zhan
authors.affiliation: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA ; Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA.
authors.forename: Guo-Wei
authors.initials: GW
authors.lastname: Wei
keywords: Discontinuous coefficients; Elliptic equations; Finite element method; Galerkin formulation; Matched interface and boundary