×

zbMATH — the first resource for mathematics

Finite element method for solving the Dirac eigenvalue problem with linear basis functions. (English) Zbl 1416.35171
Summary: In this work, we treat the spurious eigenvalues obstacle that appears in the computation of the radial Dirac eigenvalue problem using numerical methods. The treatment of the spurious solution is based on applying Petrov-Galerkin finite element method. The significance of this work is the employment of just continuous basis functions, thus the need of a continuous function which has a continuous first derivative as a basis, as in [the author, ibid. 272, 487–506 (2014; Zbl 1349.65601); et al., ibid. 236, 426–442 (2013; Zbl 1286.34119)], is no longer required. The Petrov-Galerkin finite element method for the Dirac eigenvalue problem strongly depends on a stability parameter, \(\tau\), that controls the size of the diffusion terms added to the finite element formulation for the problem. The mesh-dependent parameter \(\tau\) is derived based on the given problem with the particular basis functions.
MSC:
35P15 Estimates of eigenvalues in context of PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ackad, E.; Horbatsch, M., Numerical solution of the Dirac equation by a mapped Fourier grid method, J. Phys. A: Math. Gen., 38, 3157-3171, (2005) · Zbl 1065.81036
[2] Almanasreh, H., hp-Cloud approximation of the Dirac eigenvalue problem: the way of stability, J. Comput. Phys., 272, 487-506, (2014) · Zbl 1349.65601
[3] Almanasreh, H.; Salomonson, S.; Svanstedt, N., Stabilized finite element method for the radial Dirac equation, J. Comput. Phys., 236, 426-442, (2013) · Zbl 1286.34119
[4] Almeida, R. C.; Silva, R. S., A stable Petrov-Galerkin method for convection-dominated problems, Comput. Methods Appl. Mech. Eng., 140, (1997) · Zbl 0899.76258
[5] Dawkins, P. T.; Dunbar, S. R.; Douglass, R. W., The origin and nature of spurious eigenvalues in the spectral Tau method, J. Comput. Phys., 147, 441-462, (1998) · Zbl 0924.65077
[6] De Sampaio, P. A.B., A Petrov-Galerkin/modified operator formulation for convection-diffusion problems, Int. J. Numer. Methods Eng., 30, (1990) · Zbl 0714.76077
[7] Fischer, C. F.; Parpia, F. A., Accurate spline solutions of the radial Dirac equation, Phys. Lett. A, 179, 198-204, (1993)
[8] Fischer, C. F.; Zatsarinny, O., A B-splines Galerkin method for the Dirac equation, Comput. Phys. Commun., 180, 879-886, (2009) · Zbl 1198.81036
[9] Griesemer, M.; Lutgen, J., Accumulation of discrete eigenvalues of the radial Dirac operator, J. Funct. Anal., 162, (1999) · Zbl 0926.34074
[10] Idelsohn, S.; Nigro, N.; Storti, M.; Buscaglia, G., A Petrov-Galerkin formulation for advection-reaction-diffusion problems, Comput. Methods Appl. Mech. Eng., 136, (1996) · Zbl 0896.76042
[11] Johnson, W. R.; Blundell, S. A.; Sapirstein, J., Finite basis sets for the Dirac equation constructed from B splines, Phys. Rev. A, 37, 307-315, (1988)
[12] Lin, H.; Atluri, S. N., Meshless local Petrov-Galerkin (MLPG) method for convection-diffusion problems, Comput. Model. Eng. Sci., 1, 45-60, (2000)
[13] Lindgren, I.; Salomonson, S.; Åsén, B., The covariant-evolution-operator method in bound-state QED, Phys. Rep., 389, 161-261, (2004)
[14] Mohr, P. J.; Plunien, G.; Soff, G., QED corrections in heavy atoms, Phys. Rep., 293, 227-369, (1998)
[15] Mur, G., On the causes of spurious solutions in electromagnetics, Electromagnetics, 22, 357-367, (2002)
[16] Müller, C.; Grün, N.; Scheid, W., Finite element formulation of the Dirac equation and the problem of fermion doubling, Phys. Lett. A, 242, 245-250, (1998)
[17] Pestka, G., Spurious roots in the algebraic Dirac equation, Chem. Phys. Lett., 376, 659-661, (2003) · Zbl 1055.81016
[18] Rosenberg, L., Virtual-pair effects in atomic structure theory, Phys. Rev. A, 39, 4377-4386, (1989)
[19] Salomonson, S.; Öster, P., Relativistic all-order pair functions from a discretized single-particle Dirac Hamiltonian, Phys. Rev. A, 40, 5548-5558, (1989)
[20] Schroeder, W.; Wolf, I., The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems, IEEE Trans. Microw. Theory Tech., 42, 644-653, (1994)
[21] Shabaev, V. M., Two-time Green’s function method in quantum electrodynamics of high-Z few-electron atoms, Phys. Rep., 356, 119-228, (2002) · Zbl 1001.81090
[22] Shabaev, V. M.; Tupitsyn, I. I.; Yerokhin, V. A.; Plunien, G.; Soff, G., Dual kinetic balance approach to basis-set expansions for the Dirac equation, Phys. Rev. Lett., 93, (2004)
[23] Thaller, B., The Dirac Equation, (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0881.47021
[24] Tupitsyn, I. I.; Shabaev, V. M., Spurious states of the Dirac equation in a finite basis set, Opt. Spektrosk., 105, 203-209, (2008)
[25] Zhao, S., On the spurious solutions in the high-order finite difference methods for eigenvalue problems, Comput. Methods Appl. Mech. Eng., 196, 5031-5046, (2007) · Zbl 1173.74462
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.