## 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 $$N$$.

### MSC:

 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
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