Abstract
Two nonconforming finite elements constructed by double set parameter method are used to approximate a fourth order variational inequality with two-sided displacement obstacle. Because the exact solution does not belong to $H^{4}_{\mathrm{loc}}(\Omega)$ and each element space involves two sets of parameters, a series of novel approaches different from the exiting literature are developed in the procedure for presenting convergence analysis and deriving the optimal error estimates in broken energy norm.
Highlights
Let ⊂ R be a bounded convex polygon domain, f ∈ L ( ), ψ, ψ ∈ C ( ) ∩ C( ̄ ), ψ < ψ onand ψ < < ψ on ∂
+ wxyvxy) dx dy, (f, v) = fv dx dy, K = {v ∈ H ( ); ψ ≤ v ≤ ψ on }. It follows from the theory in [, ] that the solution of obstacle problem ( ) is uniquely determined by the following fourth order variational inequality: Find u ∈ K such that ( )
Shi and Pei Journal of Inequalities and Applications (2015) 2015:246 ity to be used in the convergence analysis of finite element methods (FEMs) [ – ]), the fourth order problem ( ) has a unique solution u belonging to H ( ) ∩ C ( ) (in general u ∈/ Hl oc( ) even for smooth data) [ – ]
Summary
Let ⊂ R be a bounded convex polygon domain, f ∈ L ( ), ψ , ψ ∈ C ( ) ∩ C( ̄ ), ψ < ψ onand ψ < < ψ on ∂. Shi and Pei Journal of Inequalities and Applications (2015) 2015:246 ity to be used in the convergence analysis of finite element methods (FEMs) [ – ]), the fourth order problem ( ) has a unique solution u belonging to H ( ) ∩ C ( ) (in general u ∈/ Hl oc( ) even for smooth data) [ – ]. In which the optimal error estimate in the energy norm with order O(h) was established by using an auxiliary obstacle problem and an enriching operator (here and later h denotes the mesh parameter). The main aim of this paper is to apply two nonconforming elements constructed by the double set parameter method to approximate problem ( ) In this situation, enriching operators meeting the requirements in [ ] become very difficult to be developed for each element space involves two sets of parameters.
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