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**On locking and robustness in the finite element method.**
*(English)*
Zbl 0763.65085

The approximation of parameter-dependent problems may lead to an unexpected convergence rate as the parameter tends to a critical value. This phenomenon has been observed in practical computations where for some choices of the discretization parameter the error may not decrease at the predicted rate for parameter values close to a critical value. The authors refer to this phenomenon as locking. To avoid such difficulties in a neighbourhood of a critical parameter \(t_ 0\) it is most natural to chose a robust method, i.e. one which is more or less uniformly convergent for all \(t\). Various robust methods have been studied in literature in the context of locking, however this article develops a systematic mathematical description that enables precise characterization of locking and robustness of an approximation to a class of variational problems. The quantities that measure locking and robustness strength are defined for a general class of variational problems and they are used in the analysis of approximation properties of several methods.

A family of parameter-dependent problems that is defined by a bilinear form \(B_ t(u,v) = a(u,v) + 1/t (Cu,Cv)\) is studied in more details, where \(a(u,v)\) is a symmetric, bilinear, \(V\)-coercive form and \(C\) is a linear form defined on certain spaces. The phenomena are discussed in a neighbourhood of the critical value of the parameter \(t_ 0=0\) for certain choices of an extension process (roughly a choice of approximation spaces) and several measures that estimate the approximation error. The locking ratio makes it possible to compare the performance of the method at a parameter value with the best possible performance for reasonable values \(t\).

Obviously, there are other reasons than locking (e.g. loss of regularity), why the accuracy of the approximation deteriorates. However, the proposed concept provides an efficient way to isolate the locking effects. A model problem is analyzed in detail and computational results are discussed. Several related problems arising in elasticity are presented together with a discussion of locking and robustness for \(h\), \(p\) and \(h\)-\(p\) approximations.

A family of parameter-dependent problems that is defined by a bilinear form \(B_ t(u,v) = a(u,v) + 1/t (Cu,Cv)\) is studied in more details, where \(a(u,v)\) is a symmetric, bilinear, \(V\)-coercive form and \(C\) is a linear form defined on certain spaces. The phenomena are discussed in a neighbourhood of the critical value of the parameter \(t_ 0=0\) for certain choices of an extension process (roughly a choice of approximation spaces) and several measures that estimate the approximation error. The locking ratio makes it possible to compare the performance of the method at a parameter value with the best possible performance for reasonable values \(t\).

Obviously, there are other reasons than locking (e.g. loss of regularity), why the accuracy of the approximation deteriorates. However, the proposed concept provides an efficient way to isolate the locking effects. A model problem is analyzed in detail and computational results are discussed. Several related problems arising in elasticity are presented together with a discussion of locking and robustness for \(h\), \(p\) and \(h\)-\(p\) approximations.

Reviewer: P.Plecháč (Bath)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35J25 | Boundary value problems for second-order elliptic equations |