Error Analysis of Symmetric Linear/Bilinear Partially Penalized Immersed Finite Element Methods for Helmholtz Interface Problems

This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise $H^2$ regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. A numerical example is conducted to validate the theoretical conclusions.

[1]  Steen Krenk,et al.  A recursive finite element technique for acoustic fields in pipes with absorption , 1988 .

[2]  C. Y. Lam,et al.  A phase-based interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber , 2017 .

[3]  Cheng Wang,et al.  An Iterative Approach for Constructing Immersed Finite Element Spaces and Applications to Interface Problems , 2018 .

[4]  Haijun Wu,et al.  Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number , 2008, SIAM J. Numer. Anal..

[5]  Marcus Sarkis,et al.  On the accuracy of finite element approximations to a class of interface problems , 2015, Math. Comput..

[7]  Ernst Rank,et al.  The finite cell method for three-dimensional problems of solid mechanics , 2008 .

[8]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[9]  Yanping Lin Qiao Zhuang Tao Lin Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods , 2019 .

[10]  Peter Hansbo,et al.  Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method , 2010 .

[11]  Zhilin Li,et al.  A Cartesian grid nonconforming immersed finite element method for planar elasticity interface problems , 2017, Comput. Math. Appl..

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

[13]  Michael B. Porter,et al.  Computational Ocean Acoustics , 1994 .

[14]  Haijun Wu,et al.  Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number , 2014, SIAM J. Numer. Anal..

[15]  A. Moiola,et al.  Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions , 2017, Mathematical Models and Methods in Applied Sciences.

[16]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .

[17]  Tao Lin,et al.  A Higher Degree Immersed Finite Element Method Based on a Cauchy Extension for Elliptic Interface Problems , 2019, SIAM J. Numer. Anal..

[18]  Jens Markus Melenk,et al.  General DG-Methods for Highly Indefinite Helmholtz Problems , 2013, J. Sci. Comput..

[19]  Qing Xia,et al.  Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D , 2018, Journal of Scientific Computing.

[20]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[21]  Théophile Chaumont-Frelet Finite element approximation of Helmholtz problems with application to seismic wave propagation , 2015 .

[22]  Guo-Wei Wei,et al.  Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces , 2007, J. Comput. Phys..

[23]  Tao Lin,et al.  Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems , 2015, SIAM J. Numer. Anal..

[24]  Wilkins Aquino,et al.  Nitsche's method for Helmholtz problems with embedded interfaces , 2017, International journal for numerical methods in engineering.

[25]  Ruchi Guo,et al.  An immersed finite element method for elliptic interface problems in three dimensions , 2019, J. Comput. Phys..

[26]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[27]  Ralf Hiptmair,et al.  PLANE WAVE DISCONTINUOUS GALERKIN METHODS: ANALYSIS OF THE h-VERSION ∗, ∗∗ , 2009 .

[28]  Charbel Farhat,et al.  A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime , 2003 .

[29]  David L. Brown A note on the numerical solution of the wave equation with piecewise smooth coefficients , 1984 .

[30]  A. H. Schatz,et al.  An observation concerning Ritz-Galerkin methods with indefinite bilinear forms , 1974 .

[31]  Haijun Wu,et al.  Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One‐Dimensional Analysis , 2012, 1211.1424.

[32]  R. B. Kellogg,et al.  A scattering problem for the Helmholtz equation , 1979 .

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

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

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

[36]  Luke Swift,et al.  Geometrically unfitted finite element methods for the Helmholtz equation , 2018 .

[37]  Hélène Barucq,et al.  Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation , 2016, Math. Comput..

[38]  Ernst Rank,et al.  Theoretical and Numerical Investigation of the Finite Cell Method , 2015, Journal of Scientific Computing.

[39]  A. K. Aziz,et al.  On the Numerical Solutions of Helmholtz’s Equation by the Finite Element Method , 1980 .

[40]  Christoph Lehrenfeld,et al.  A note on the stability parameter in Nitsche's method for unfitted boundary value problems , 2017, Comput. Math. Appl..

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

[42]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[43]  Q. Zou,et al.  ANALYSIS OF A SPECIAL IMMERSED FINITE VOLUME METHOD FOR ELLIPTIC INTERFACE PROBLEMS , 2019 .

[44]  I. Babuska,et al.  Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM☆ , 1995 .

[45]  Reflection and transmission of plane waves at an interface between two fluids , 2007 .

[46]  Jim Douglas,et al.  APPROXIMATION OF SCALAR WAVES IN THE SPACE-FREQUENCY DOMAIN , 1994 .

[47]  Heinz-Otto Kreiss,et al.  An Embedded Boundary Method for the Wave Equation with Discontinuous Coefficients , 2005, SIAM J. Sci. Comput..