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A new cloud-based $$hp$$ finite element method. (English) Zbl 0956.74062
Conclusions: We explore the use of lower-order finite element approximations to generate partitions of unity on which hierarchical $$hp$$-cloud approximations can be constructed. The resulting methodology has a number of useful features. Among these are that non-uniform $$hp$$-meshing with variable and hierarchical order $$p$$ over clouds can be easily generated. The spectral convergence of $$p$$- and $$hp$$-methods is retained and the method is very robust, the accuracy being quite insensitive to mesh distortion. Also, by building cloud approximations on finite element meshes, Dirichlet boundary conditions are easily handled. The $$hp$$-convergence properties seem to differ from traditional $$p$$-version elements, but exponential convergence is attained. Applications to problems with singularities are easily handled using cloud schemes. In all, this hybrid finite-element/cloud methodology appears to have a number of useful and attractive features that could prove to be important in broad engineering applications.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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##### References:
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