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On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. (English) Zbl 1069.65116
Summary: This paper proposes a weak patch test that improves both the patch test and the generalized patch test. Under a weak superapproximation assumption, which can be obtained from certain approximation and weak continuity property, it is shown that the weak patch test is equivalent to the generalized patch test. Thus if a nonconforming finite element passes the patch test and satisfies some approximation and weak continuity property, its convergence is then guaranteed. Furthermore, the consistency term of such a nonconforming element can be proved to be of order $O\left(h\right)$. It is shown that if a nonconforming element does not pass the patch test and a condition relative to subdividing a domain by the element is true, then one can find a family of triangulations for which the element is divergent.
##### MSC:
 65N12 Stability and convergence of numerical methods (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems 65N50 Mesh generation and refinement (BVP of PDE)