Interpolation Based Local Postprocessing for Adaptive Finite Element Approximations in Electronic Structure Calculations

In this paper, we propose an interpolation based local postprocessing approach for finite element electronic structure calculations over locally refined hexahedral finite element meshes. It is shown that our approach is very efficient in finite element approximations of ground state energies.

[1]  Sullivan,et al.  Real-space multigrid-based approach to large-scale electronic structure calculations. , 1996, Physical review. B, Condensed matter.

[2]  Percy Deift,et al.  Review: Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators , 1985 .

[3]  I. Babuska,et al.  Finite element-galerkin approximation of the eigenvalues and Eigenvectors of selfadjoint problems , 1989 .

[4]  Barry Simon,et al.  Schrödinger operators in the twentieth century , 2000 .

[5]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[6]  T. Beck Real-space mesh techniques in density-functional theory , 2000, cond-mat/0006239.

[7]  Harry Yserentant,et al.  The Electronic Schrödinger Equation , 2010 .

[8]  Richard M. Martin Electronic Structure: Frontmatter , 2004 .

[9]  P. Pulay Convergence acceleration of iterative sequences. the case of scf iteration , 1980 .

[10]  Jinchao Xu,et al.  Local and Parallel Finite Element Algorithms for Eigenvalue Problems , 2002 .

[11]  Aihui Zhou,et al.  A Defect Correction Scheme for Finite Element Eigenvalues with Applications to Quantum Chemistry , 2006, SIAM J. Sci. Comput..

[12]  R. Martin,et al.  Electronic Structure: Basic Theory and Practical Methods , 2004 .

[13]  Jinchao Xu,et al.  Local and parallel finite element algorithms based on two-grid discretizations , 2000, Math. Comput..

[14]  P. Pulay Improved SCF convergence acceleration , 1982 .

[15]  G. Burton Sobolev Spaces , 2013 .

[16]  I. Babuška,et al.  Corrigendum: “Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems” [Math. Comp. 52 (1989), no. 186, 275–297; MR0962210 (89k:65132)] , 1994 .

[17]  Jinchao Xu,et al.  A two-grid discretization scheme for eigenvalue problems , 2001, Math. Comput..

[18]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[19]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[20]  John R. Brauer What every engineer should know about finite element analysis , 1995 .

[21]  Ping Wang,et al.  On the Monotonicity of (k;g,h)-graphs , 2002 .

[22]  S. Goedecker Linear scaling methods for the solution of schrödinger's equation , 2003 .

[23]  M. Tsukada,et al.  Electronic-structure calculations based on the finite-element method. , 1995, Physical review. B, Condensed matter.

[24]  Gong,et al.  FINITE ELEMENT APPROXIMATIONS FOR SCHR ¨ ODINGER EQUATIONS WITH APPLICATIONS TO ELECTRONIC STRUCTURE COMPUTATIONS * , 2008 .

[25]  A. H. Schatz,et al.  Interior maximum-norm estimates for finite element methods, part II , 1995 .

[26]  Aihui Zhou,et al.  Three-scale finite element eigenvalue discretizations , 2008 .

[27]  J. Pask,et al.  Finite element methods in ab initio electronic structure calculations , 2005 .

[28]  Aihui Zhou,et al.  Three-Scale Finite Element Discretizations for Quantum Eigenvalue Problems , 2007, SIAM J. Numer. Anal..

[29]  Xiaoying Dai,et al.  A LOCAL COMPUTATIONAL SCHEME FOR HIGHER ORDER FINITE ELEMENT EIGENVALUE APPROXIMATIONS , 2008 .

[30]  Kohn,et al.  Density functional and density matrix method scaling linearly with the number of atoms. , 1996, Physical review letters.

[31]  Lihua Shen,et al.  Finite element method for solving Kohn-Sham equations based on self-adaptive tetrahedral mesh , 2008 .