Complementary Solutions of Nitsche’s Method

Abstract

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.

Appendix: The Irons–Guyan Reduced Nitsche Formulation

Consider the generalized algebraic eigenvalue problem emanating from discretization of the Nitsche formulation in partitioned form

$$\begin{aligned} \left( \left[ \begin{array}{ll} {\mathbf {K}}_{\mathrm {ss}}&{}\quad {\mathbf {K}}_{\mathrm {sn}}\\ {\mathbf {K}}_{\mathrm {ns}}&{}\quad {\mathbf {K}}_{\mathrm {nn}}^{\alpha }\end{array}\right] - \lambda \left[ \begin{array}{ll} {\mathbf {M}}_{\mathrm {ss}}&{} {\mathbf {M}}_{\mathrm {sn}}\\ {\mathbf {M}}_{\mathrm {ns}}&{} {\mathbf {M}}_{\mathrm {nn}}\end{array}\right] \right) \left\{ \begin{array}{l} {\varvec{\psi }}_\mathrm {s}\\ {\varvec{\psi }}_\mathrm {n}\end{array}\right\} = \mathbf {0} \end{aligned}$$

(46)

The degrees of freedom of the eigenvector in ${\varvec{\psi }}_\mathrm {s}$, corresponding to those of the underlying kinematically admissible discretization, are to be retained in the reduced problem. Those in ${\varvec{\psi }}_\mathrm {n}$, corresponding to the additional degrees of freedom in the discrete Nitsche approach, are to be removed. The ${\mathbf {K}}$ matrices arise from the discretization of a(w, u) in (2), and the ${\mathbf {M}}$ terms from (w, u). The diagonal blocks are symmetric, and each coupling term is the transpose of the corresponding off-diagonal block. The stabilization parameter appears only in the relatively small matrix ${\mathbf {K}}_{\mathrm {nn}}^{\alpha }$. The reduced problem is

$$\begin{aligned} \left( {\mathbf {K}}^* - \lambda {\mathbf {M}}^* \right) {\varvec{\psi }}_\mathrm {s}= \mathbf {0} \end{aligned}$$

(47)

The Irons–Guyan reduced matrices are

$$\begin{aligned} {\mathbf {K}}^*= & {} {\mathbf {K}}_{\mathrm {ss}}- {\mathbf {K}}_{\mathrm {sn}}\left( {\mathbf {K}}_{\mathrm {nn}}^{\alpha }\right) ^{-1}{\mathbf {K}}_{\mathrm {ns}}\end{aligned}$$

(48)

$$\begin{aligned} {\mathbf {M}}^*= & {} {\mathbf {M}}_{\mathrm {ss}}- {\mathbf {M}}_{\mathrm {sn}}\left( {\mathbf {K}}_{\mathrm {nn}}^{\alpha }\right) ^{-1}{\mathbf {K}}_{\mathrm {ns}}- {\mathbf {K}}_{\mathrm {sn}}\left( {\mathbf {K}}_{\mathrm {nn}}^{\alpha }\right) ^{-1}\left( {\mathbf {M}}_{\mathrm {ns}}- {\mathbf {M}}_{\mathrm {nn}}\left( {\mathbf {K}}_{\mathrm {nn}}^{\alpha }\right) ^{-1}{\mathbf {K}}_{\mathrm {ns}}\right) \end{aligned}$$

(49)

Diagonal scaling may be used to precondition the procedure. In this case, every appearance of $\left( {\mathbf {K}}_{\mathrm {nn}}^{\alpha }\right) ^{-1}$ is replaced by ${\mathbf {D}}\left( {\mathbf {D}}\mathbf {K}_{\mathrm {nn}}^\alpha {\mathbf {D}}\right) ^{-1}{\mathbf {D}}$. Here, ${\mathbf {D}}= 1/\sqrt{\mathrm {diag} \mathbf {K}_{\mathrm {nn}}^\alpha }$.