Solving Eigenvalue Problems in a Discontinuous Approximation Space by Patch Reconstruction

We adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order pro...

[1]  Jun Hu,et al.  The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods , 2011, 1112.1145.

[2]  Gianmarco Manzini,et al.  Convergence of the mimetic finite difference method for eigenvalue problems in mixed form , 2011 .

[3]  V. G. Sigillito,et al.  Eigenvalues of the Laplacian in Two Dimensions , 1984 .

[4]  Ruo Li,et al.  A discontinuous Galerkin method for Stokes equation by divergence‐free patch reconstruction , 2018, Numerical Methods for Partial Differential Equations.

[5]  Ruo Li,et al.  An Efficient High Order Heterogeneous Multiscale Method for Elliptic Problems , 2012, Multiscale Model. Simul..

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

[7]  Junping Wang,et al.  Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes , 2012, J. Comput. Appl. Math..

[8]  A. Buffa,et al.  Discontinuous Galerkin approximation of the Laplace eigenproblem , 2006 .

[9]  G. Paulino,et al.  PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab , 2012 .

[10]  Susanne C. Brenner,et al.  C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems , 2015 .

[11]  Gianmarco Manzini,et al.  The Mimetic Finite Difference Method for Elliptic Problems , 2014 .

[12]  Jun Hu,et al.  Guaranteed Lower Bounds for Eigenvalues of Elliptic Operators , 2016, J. Sci. Comput..

[13]  J. Hesthaven,et al.  High–order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[15]  T. Dupont,et al.  Polynomial approximation of functions in Sobolev spaces , 1980 .

[16]  Ruo Li,et al.  An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element , 2018, Journal of Scientific Computing.

[17]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[18]  Annalisa Buffa,et al.  Mimetic finite differences for elliptic problems , 2009 .

[19]  Susanne C. Brenner,et al.  A Locally Divergence-Free Interior Penalty Method for Two-Dimensional Curl-Curl Problems , 2008, SIAM J. Numer. Anal..

[20]  P. Ming,et al.  A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem , 2017, Journal of Computational Mathematics.

[21]  Ricardo G. Durán,et al.  Mass‐lumping or not mass‐lumping for eigenvalue problems , 2003 .

[22]  George E. Forsythe,et al.  Asymptotic lower bounds for the frequencies of certain polygonal membranes. , 1954 .

[23]  Ruo Li,et al.  A finite element method by patch reconstruction for the Stokes problem using mixed formulations , 2018, J. Comput. Appl. Math..

[24]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[25]  Zhimin Zhang,et al.  How Many Numerical Eigenvalues Can We Trust? , 2013, J. Sci. Comput..

[26]  Mark Embree,et al.  The role of the penalty in the local discontinuous Galerkin method for Maxwell’s eigenvalue problem , 2006 .

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

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

[29]  Stefano Giani,et al.  An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods , 2018, Appl. Math. Comput..

[30]  T. Hughes,et al.  Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity , 2002 .

[31]  D. W. Scharpf,et al.  The TUBA Family of Plate Elements for the Matrix Displacement Method , 1968, The Aeronautical Journal (1968).

[32]  Jun Hu,et al.  THE LOWER APPROXIMATION OF EIGENVALUE BY LUMPED MASS FINITE ELEMENT METHOD ∗1) , 2004 .

[33]  Xuefeng Liu,et al.  Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape , 2012, SIAM J. Numer. Anal..

[34]  Jun Hu,et al.  The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators , 2014, J. Sci. Comput..

[35]  S. C. Brenner,et al.  C 0 Interior Penalty Methods , 2011 .

[36]  JunHu,et al.  THE LOWER APPROXIMATION OF EIGENVALUE BY LUMPED MASS FINITE ELEMENT METHOD , 2004 .

[37]  R. Durán,et al.  ASYMPTOTIC LOWER BOUNDS FOR EIGENVALUES BY NONCONFORMING FINITE ELEMENT METHODS , 2004 .