Journal of Scientific Computing

, Volume 81, Issue 3, pp 1472–1492 | Cite as

Complementary Solutions of Nitsche’s Method

  • Isaac HarariEmail author
  • Uri Albocher


Embedded methods that are based on Nitsche’s approach can facilitate the task of mesh generation in many configurations. The basic workings of the method are well understood, in terms of a bound on the stabilization parameter. However, its spectral behavior has not been explored in depth. In addition to the eigenpairs which approximate the exact ones, as in the standard formulation, Nitsche’s method gives rise to mesh-dependent complementary pairs. The dependence of the eigenvalues on the Nitsche parameter is related to a boundary quotient of the eigenfunctions, explaining the manner in which stabilization engenders coercivity without degrading the accuracy of the discrete eigenpairs. The boundary quotient proves to be useful for separating the two types of solutions. The quotient space is handy for determining the number of eigenpairs and complementary pairs. The complementary solutions approximate functions in the orthogonal complement of the kinematically admissible subspace. A global result for errors in the Galerkin approximation of the eigenvalue problem that pertains to all modes of the discretization, is extended to the Nitsche formulation. Numerical studies on non-conforming aligned meshes confirm the dependence of the eigenvalues on the parameter, in line with the corresponding boundary quotients. The spectrum of a reduced system obtained by algebraic elimination is free of complementary solutions, warranting its use in the solution of boundary-value problems. The reduced system offers an incompatible discretization of eigenvalue problems that is suitable for engineering applications. Using Irons–Guyan reduction yields a spectrum that is virtually insensitive to stabilization, with high accuracy in both eigenvalues and eigenfunctions.


Incompatible discretization Nitsche’s method Spectral behavior Eigenvalue problem 



The authors thank Adrian Lew for helpful discussions on boundary locking. This research was supported by the Israel Science Foundation (Grant No. 269/17), and by the Diane and Arthur B. Belfer Chair in Mechanics and Biomechanics.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of EngineeringTel Aviv UniversityTel AvivIsrael
  2. 2.Afeka, Tel Aviv Academic College of EngineeringTel AvivIsrael

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