Adaptive dimensional reduction numerical solution of monotone quasilinear boundary value problems. (English) Zbl 0778.73069

The author continues the development of a method for dimensional reduction of problems arising in structural analysis of “thin” elastic structures. When a problem is to be solved over the rectangle \([0,1]\times[-d,d]\), with small \(d\), then the author’s approach is to approximate the solution as a sum \(\sum^ N_ 0c_ j(\xi)\psi_ j(\eta)\) with \(\xi \in[0,1]\) and \(\eta d=y \in[-d,d]\).
The results in this paper rely upon and extend those in [S. Jensen, I. Babuška, SIAM J. Numer. Anal. 25, No. 3, 644-669 (1988; Zbl 0656.73040)]. The extension concentrates on proving that it is possible to choose different orders \(N\) over different subintervals of [0,1], exhibiting error estimators which make such choices practical, and presenting numerical results supporting the theory.
The author is interested in the problems of the following class. Find \(u \in W\) such that \(Au(v) = G(v)\) for \(\forall\) \(v \in W\), where \(Au(v)=\int_ \Omega F(| \nabla_ d |^ 2) \nabla_ du \cdot \nabla_ dvdd \xi d \eta\), \(G(v)=d^{1-\mu}\int^ 1_ 0\beta(\xi)[v(\xi,1)+v(\xi,-1)]d\xi\), \(\nabla_ d=\left({\partial \over \partial \xi},{1\over d}{\partial \over \partial \eta} \right)\) and \(F(t)=1+t^ n\). A key error estimator is \(Est=\left \|{1\over d}{\partial e \over \partial \eta} \right\|_ 2\), where \(e\) is a solution of \(\int^ 1_{-1}\int^ 1_ 0{1 \over d}{\partial e \over \partial \eta}{1\over d}{\partial v \over \partial \eta}dd \xi d \eta\). This estimator can also be defined over subintervals of [0,1].
A main result is the presentation of several heuristics for choosing subintervals with different orders \(N\) and the proof that they are optimal with respect to convergence rate for large \(N\).


74S05 Finite element methods applied to problems in solid mechanics
74K99 Thin bodies, structures
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N45 Method of contraction of the boundary for boundary value problems involving PDEs


Zbl 0656.73040
Full Text: DOI