2015, 20(5): 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential

1. 

Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

2. 

Fariborz Maseeh Department of Mathematics & Statistics, Portland State University, Portland, OR 97201, United States

Received  August 2013 Revised  January 2015 Published  May 2015

We develop an a posteriori error estimate of hierarchical type for Dirichlet eigenvalue problems of the form $(-\Delta+(c/r)^2)\psi=\lambda \psi$ on bounded domains $\Omega$, where $r$ is the distance to the origin, which is assumed to be in $\overline\Omega$. This error estimate is proven to be asymptotically identical to the eigenvalue approximation error on a family of geometrically-graded meshes. Numerical experiments demonstrate this asymptotic exactness in practice.
Citation: Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
References:
[1]

T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges,, Math. Methods Appl. Sci., 21 (1998), 519. doi: 10.1002/(SICI)1099-1476(199804)21:6<519::AID-MMA962>3.0.CO;2-R.

[2]

I. Babuška, R. B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements,, Numer. Math., 33 (1979), 447. doi: 10.1007/BF01399326.

[3]

I. Babuška and J. Osborn, Eigenvalue problems,, In Handbook of numerical analysis, (1991), 641.

[4]

C. Băcuţă, V. Nistor and L. T. Zikatanov, Improving the rate of convergence of 'high order finite elements' on polygons and domains with cusps,, Numer. Math., 100 (2005), 165. doi: 10.1007/s00211-005-0588-3.

[5]

R. E. Bank, PLTMG: A software package for solving elliptic partial differential equations. Users' Guide 10.0,, Technical report, (2007).

[6]

R. E. Bank, L. Grubišić and J. S. Ovall, A framework for robust eigenvalue and eigenvector error estimation and ritz value convergence enhancement,, Applied Numer. Math., 66 (2013), 1. doi: 10.1016/j.apnum.2012.11.004.

[7]

H. Chen, L. He and A. Zhou, Finite element approximations of nonlinear eigenvalue problems in quantum physics,, Comput. Methods Appl. Mech. Engrg., 200 (2011), 1846. doi: 10.1016/j.cma.2011.02.008.

[8]

M. Dauge, Elliptic Boundary Value Problems on Corner Domains, volume 1341 of Lecture Notes in Mathematics,, Springer-Verlag, (1988).

[9]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, volume 159 of {Applied Mathematical Sciences},, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-4355-5.

[10]

V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential,, J. Eur. Math. Soc. (JEMS), 13 (2011), 119. doi: 10.4171/JEMS/246.

[11]

V. Felli, E. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity,, Discrete Contin. Dyn. Syst., 21 (2008), 91. doi: 10.3934/dcds.2008.21.91.

[12]

S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergaard Sørensen, Analytic structure of solutions to multiconfiguration equations,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/31/315208.

[13]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics,, Pitman (Advanced Publishing Program), (1985).

[14]

P. Grisvard, Singularities in Boundary Value Problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics],, Masson, (1992).

[15]

E. Hunsicker, H. Li, V. Nistor and U. Ville, Analysis of Schrödinger operators with inverse square potentials I: Regularity results in 3D,, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55 (2012), 157.

[16]

V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points,, Trudy Moskov. Mat. Obšč., 16 (1967), 209.

[17]

V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, volume 52 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997).

[18]

V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, volume 85 of Mathematical Surveys and Monographs,, American Mathematical Society, (2001).

[19]

R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide, volume 6 of Software, Environments, and Tools,, Society for Industrial and Applied Mathematics (SIAM), (1998). doi: 10.1137/1.9780898719628.

[20]

H. Li, A. Mazzucato and V. Nistor, Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains,, Electron. Trans. Numer. Anal., 37 (2010), 41.

[21]

H. Li and V. Nistor, Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM,, J. Comput. Appl. Math., 224 (2009), 320. doi: 10.1016/j.cam.2008.05.009.

[22]

H. Li and J. S. Ovall, A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential,, Numer. Math, 128 (2014), 707. doi: 10.1007/s00211-014-0628-y.

[23]

S. Moroz and R. Schmidt, Nonrelativistic inverse square potential, scale anomaly, and complex extension,, Annals of Physics, 325 (2010), 491. doi: 10.1016/j.aop.2009.10.002.

[24]

A. Naga and Z. Zhang, Function value recovery and its application in eigenvalue problems,, SIAM J. Numer. Anal., 50 (2012), 272. doi: 10.1137/100797709.

[25]

G. Strang and G. J. Fix, An Analysis of the Finite Element Method,, Prentice-Hall Inc., (1973).

[26]

L. N. Trefethen and T. Betcke, Computed eigenmodes of planar regions,, In Recent advances in differential equations and mathematical physics, (2006), 297. doi: 10.1090/conm/412/07783.

[27]

N. M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems,, J. Math. Mech., 13 (1964), 549.

[28]

H. Wu and D. W. L. Sprung, Inverse-square potential and the quantum vortex,, Phys. Rev. A, 49 (1994), 4305. doi: 10.1103/PhysRevA.49.4305.

show all references

References:
[1]

T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges,, Math. Methods Appl. Sci., 21 (1998), 519. doi: 10.1002/(SICI)1099-1476(199804)21:6<519::AID-MMA962>3.0.CO;2-R.

[2]

I. Babuška, R. B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements,, Numer. Math., 33 (1979), 447. doi: 10.1007/BF01399326.

[3]

I. Babuška and J. Osborn, Eigenvalue problems,, In Handbook of numerical analysis, (1991), 641.

[4]

C. Băcuţă, V. Nistor and L. T. Zikatanov, Improving the rate of convergence of 'high order finite elements' on polygons and domains with cusps,, Numer. Math., 100 (2005), 165. doi: 10.1007/s00211-005-0588-3.

[5]

R. E. Bank, PLTMG: A software package for solving elliptic partial differential equations. Users' Guide 10.0,, Technical report, (2007).

[6]

R. E. Bank, L. Grubišić and J. S. Ovall, A framework for robust eigenvalue and eigenvector error estimation and ritz value convergence enhancement,, Applied Numer. Math., 66 (2013), 1. doi: 10.1016/j.apnum.2012.11.004.

[7]

H. Chen, L. He and A. Zhou, Finite element approximations of nonlinear eigenvalue problems in quantum physics,, Comput. Methods Appl. Mech. Engrg., 200 (2011), 1846. doi: 10.1016/j.cma.2011.02.008.

[8]

M. Dauge, Elliptic Boundary Value Problems on Corner Domains, volume 1341 of Lecture Notes in Mathematics,, Springer-Verlag, (1988).

[9]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, volume 159 of {Applied Mathematical Sciences},, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-4355-5.

[10]

V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential,, J. Eur. Math. Soc. (JEMS), 13 (2011), 119. doi: 10.4171/JEMS/246.

[11]

V. Felli, E. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity,, Discrete Contin. Dyn. Syst., 21 (2008), 91. doi: 10.3934/dcds.2008.21.91.

[12]

S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergaard Sørensen, Analytic structure of solutions to multiconfiguration equations,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/31/315208.

[13]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics,, Pitman (Advanced Publishing Program), (1985).

[14]

P. Grisvard, Singularities in Boundary Value Problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics],, Masson, (1992).

[15]

E. Hunsicker, H. Li, V. Nistor and U. Ville, Analysis of Schrödinger operators with inverse square potentials I: Regularity results in 3D,, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55 (2012), 157.

[16]

V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points,, Trudy Moskov. Mat. Obšč., 16 (1967), 209.

[17]

V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, volume 52 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997).

[18]

V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, volume 85 of Mathematical Surveys and Monographs,, American Mathematical Society, (2001).

[19]

R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide, volume 6 of Software, Environments, and Tools,, Society for Industrial and Applied Mathematics (SIAM), (1998). doi: 10.1137/1.9780898719628.

[20]

H. Li, A. Mazzucato and V. Nistor, Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains,, Electron. Trans. Numer. Anal., 37 (2010), 41.

[21]

H. Li and V. Nistor, Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM,, J. Comput. Appl. Math., 224 (2009), 320. doi: 10.1016/j.cam.2008.05.009.

[22]

H. Li and J. S. Ovall, A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential,, Numer. Math, 128 (2014), 707. doi: 10.1007/s00211-014-0628-y.

[23]

S. Moroz and R. Schmidt, Nonrelativistic inverse square potential, scale anomaly, and complex extension,, Annals of Physics, 325 (2010), 491. doi: 10.1016/j.aop.2009.10.002.

[24]

A. Naga and Z. Zhang, Function value recovery and its application in eigenvalue problems,, SIAM J. Numer. Anal., 50 (2012), 272. doi: 10.1137/100797709.

[25]

G. Strang and G. J. Fix, An Analysis of the Finite Element Method,, Prentice-Hall Inc., (1973).

[26]

L. N. Trefethen and T. Betcke, Computed eigenmodes of planar regions,, In Recent advances in differential equations and mathematical physics, (2006), 297. doi: 10.1090/conm/412/07783.

[27]

N. M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems,, J. Math. Mech., 13 (1964), 549.

[28]

H. Wu and D. W. L. Sprung, Inverse-square potential and the quantum vortex,, Phys. Rev. A, 49 (1994), 4305. doi: 10.1103/PhysRevA.49.4305.

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