A selective immersed discontinuous Galerkin method for elliptic interface problems

Abstract

This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.

Three‐dimensional immersed finite‐element method for anisotropic magnetostatic/electrostatic interface problems with nonhomogeneous flux jump

“…This minimum modification of the finite‐element method leads to a minimum and convenient modification of the traditional finite element code to fit the new needs of solving interface problems on Cartesian meshes. The IFE spaces are also extended to the discontinuous Galerkin method and the finite volume element method . For the analysis of the IFE methods, see References and references therein.…”

Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods

“…With Hesie-Clough-Tocher type [11,20] macro polynomials constructed according to the interface jump conditions as the local shape functions, IFE methods allow the interface to split the interior of the elements in a mesh; therefore, IFE methods are interface-independent methods that can use highly structured Cartesian meshes for problems with non-trivial interfaces, see [41,42,51] for some IFE spaces based on triangular Cartesian meshes and [26,29,31,43] for some IFE spaces based rectangular Catesian meshes. Partially penalized IFE (PPIFE) and DGIFE methods [27,30,31,44,45,56] have been developed for elliptic interface problems, and the related extensions to higher degree IFE methods are presented in [1][2][3]48].…”

An immersed weak Galerkin method for elliptic interface problems

“…The IFEM was first developed for elliptic interface problems [2,8,23,26,27] and was recently applied to other interface model problems such as elasticity system [33,30], Stokes flow [1], parabolic moving interface problems [17,28], etc. Recently, this immersed idea has also been used in various numerical algorithms other than classical conforming FEM, such as nonconforming IFEM [21,31], immersed Petrov-Galerkin methods [18,19], immersed discontinuous Galerkin methods [16,32], and immersed finite volume methods [9,15].…”

“…One apparent advantage of our IWG method over standard WG method is that it can be applied on unfitted meshes such as Cartesian meshes for solving elliptic interface problems. Comparing with the immersed IPDG methods [16,32], the matrix assembling in the IWG method assembles is more efficiently because all computation can be done locally within an element without exchange information from neighboring elements.…”