Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient

This paper applies bilinear immersed finite elements (IFEs) in the interior penalty discontinuous Galerkin (DG) methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bilinear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.

[1]  I. Babuska,et al.  Finite Element Methods for the Solution of Problems with Rough Input Data. , 1985 .

[2]  Weiwei Sun,et al.  Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems , 2007 .

[3]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[4]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[5]  Hong Wang,et al.  A Family of Characteristic Discontinuous Galerkin Methods for Transient Advection-Diffusion Equations and Their Optimal-Order L^2 Error Estimates , 2009 .

[6]  Francis X. Giraldo,et al.  A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier-Stokes Equations in Nonhydrostatic Mesoscale Modeling , 2009, SIAM J. Sci. Comput..

[7]  Per-Olof Persson,et al.  The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems , 2007, SIAM J. Sci. Comput..

[8]  Michael Dumbser,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[9]  Dong Liang,et al.  Numerical analysis to discontinuous Galerkin methods for the age structured population model of marine invertebrates , 2009 .

[10]  Johnny Guzmán.  Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem , 2008, Math. Comput..

[11]  Christian Rohde,et al.  Local Discontinuous-Galerkin Schemes for Model Problems in Phase Transition Theory , 2008 .

[12]  Zhiliang Xu,et al.  Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells , 2009, J. Comput. Phys..

[13]  Zhilin Li,et al.  An immersed finite element space and its approximation capability , 2004 .

[14]  Thomas Y. Hou,et al.  GLOBAL WELL-POSEDNESS OF THE VISCOUS BOUSSINESQ EQUATIONS , 2004 .

[15]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[16]  Ivo Babuska,et al.  The finite element method for elliptic equations with discontinuous coefficients , 1970, Computing.

[17]  Derek M. Causon,et al.  Developments in Cartesian cut cell methods , 2003, Math. Comput. Simul..

[18]  Weiwei Sun,et al.  Quadratic immersed finite element spaces and their approximation capabilities , 2006, Adv. Comput. Math..

[19]  Mary F. Wheeler,et al.  Coupling Discontinuous Galerkin and Mixed Finite Element Discretizations using Mortar Finite Elements , 2008, SIAM J. Numer. Anal..

[20]  L. M. Delves,et al.  An Implicit Matching Principle for Global Element Calculations , 1979 .

[21]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[22]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[23]  M. Wheeler.  An Elliptic Collocation-Finite Element Method with Interior Penalties , 1978 .

[24]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[25]  Kai Fan,et al.  A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions , 2008, J. Comput. Phys..

[26]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .

[27]  Yanping Lin,et al.  A rectangular immersed finite element space for interface problems , 2001 .

[28]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[29]  Ralf Hartmann,et al.  Smoothed Aggregation Multigrid for the Discontinuous Galerkin Method , 2009, SIAM J. Sci. Comput..

[30]  Kaixin Wang.  A uniform optimal‐order estimate for an Eulerian‐Lagrangian discontinuous Galerkin method for transient advection–diffusion equations , 2009 .

[31]  Yan Xu,et al.  Local Discontinuous Galerkin Method for the Hunter--Saxton Equation and Its Zero-Viscosity and Zero-Dispersion Limits , 2008, SIAM J. Sci. Comput..

[32]  Chi-Wang Shu,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[33]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[34]  Stefan A. Sauter,et al.  Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients , 2006, Computing.

[35]  Jean-François Remacle,et al.  High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations , 2009, J. Comput. Phys..

[36]  Dennis W. Hewett,et al.  The Embedded Curved Boundary Method for Orthogonal Simulation Meshes , 1997 .

[37]  Tao Lin,et al.  New Cartesian grid methods for interface problems using the finite element formulation , 2003, Numerische Mathematik.

[38]  Chi-Wang Shu,et al.  The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode , 2009, J. Sci. Comput..

[39]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[40]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[41]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[42]  I. Babuska,et al.  Nonconforming Elements in the Finite Element Method with Penalty , 1973 .

[43]  Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptical Problems Part I. Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptic Problems , 1999 .

[44]  Slimane Adjerid,et al.  HIGHER-ORDER IMMERSED DISCONTINUOUS GALERKIN METHODS , 2007 .

[45]  Wenbin Liu,et al.  DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD WITH INTERIOR PENALTIES FOR CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEM , 2009 .

[46]  Xiaoming He,et al.  Approximation capability of a bilinear immersed finite element space , 2008 .

[47]  Bo Li,et al.  Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions , 2007, SIAM J. Numer. Anal..

[48]  J. K. Djoko.  Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds , 2008 .

[49]  John E. Osborn,et al.  Can a finite element method perform arbitrarily badly? , 2000, Math. Comput..

[50]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[51]  D. Schötzau,et al.  Interior penalty discontinuous Galerkin method for Maxwell's equations , 2007 .

[52]  B. Heinrich.  Finite Difference Methods on Irregular Networks , 1987 .

[53]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[54]  Danping Yang,et al.  Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations , 2008 .

[55]  Jianxian Qiu,et al.  Simulations of compressible two-medium flow by Runge-Kutta discontinuous Galerkin methods with the ghost fluid method , 2008 .

[56]  Bo Dong,et al.  Analysis of a Local Discontinuous Galerkin Method for Linear Time-Dependent Fourth-Order Problems , 2009, SIAM J. Numer. Anal..

[57]  Wei Cai,et al.  A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides , 2008, J. Comput. Phys..

[58]  Amiya K. Pani,et al.  Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems , 2009 .

[59]  Rolf Rannacher,et al.  Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems , 2002, J. Num. Math..

[60]  Slimane Adjerid,et al.  A p-th degree immersed finite element for boundary value problems with discontinuous coefficients , 2009 .

[61]  Kanschat Guido.  Block Preconditioners for LDG Discretizations of Linear Incompressible Flow Problems , 2005 .

[62]  Zhangxin Chen,et al.  Stability and convergence of mixed discontinuous finite element methods for second-order differential problems , 2003, J. Num. Math..

[63]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

[64]  Zhilin Li.  The immersed interface method using a finite element formulation , 1998 .

[65]  I. Babuska.  The Finite Element Method with Penalty , 1973 .

[66]  Xiaoming He,et al.  A Bilinear Immersed Finite Volume Element Method For the Diffusion Equation with Discontinuous Coefficient , 2009 .

[67]  R. Kafafy,et al.  Three‐dimensional immersed finite element methods for electric field simulation in composite materials , 2005 .

[68]  Jaap J. W. van der Vegt,et al.  A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids , 2008, J. Comput. Phys..

[69]  Zhangxin Chen,et al.  Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case , 2003 .

[70]  T. F. Russell,et al.  Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis , 1994 .

[71]  Jianxian Qiu,et al.  Adaptive Runge-Kutta discontinuous Galerkin methods using different indicators: One-dimensional case , 2009, J. Comput. Phys..

[72]  James H. Bramble,et al.  A finite element method for interface problems in domains with smooth boundaries and interfaces , 1996, Adv. Comput. Math..

[73]  Bernardo Cockburn,et al.  Error Estimates for the Runge-Kutta Discontinuous Galerkin Method for the Transport Equation with Discontinuous Initial Data , 2008, SIAM J. Numer. Anal..

[74]  Shan Zhao,et al.  High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources , 2006, J. Comput. Phys..

[75]  Tzin Shaun Wang.  A Hermite Cubic Immersed Finite Element Space for Beam Design Problems , 2005 .

[76]  Zhilin Li,et al.  The immersed finite volume element methods for the elliptic interface problems , 1999 .

[77]  Z. Chen,et al.  On the relationship of various discontinuous finite element methods for second-order elliptic equations , 2001, J. Num. Math..

[78]  Xiaoming He.  Bilinear Immersed Finite Elements for Interface Problems , 2009 .

[79]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .

[80]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[81]  Thomas J. R. Hughes,et al.  A comparison of discontinuous and continuous Galerkin methods bases on error estimates, conservation, robustness and efficiency , 2000 .

[82]  Benjamin Stamm,et al.  Local discontinuous Galerkin method for diffusion equations with reduced stabilization , 2009 .

[83]  Xiaoming He,et al.  Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions , 2011 .